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style='mso-fareast-font-family:"Times New Roman"'> <hr size=5 width="100%" noshade style='color:#C80000' align=center> </span></div> <p>&nbsp;</p> <p class=MsoNormal align=center style='text-align:center'><span style='mso-fareast-font-family:"Times New Roman";mso-no-proof:yes'><img width=356 height=195 id="_x0000_i1041" src="../EMAT6680Logo.jpg"></span><span style='mso-fareast-font-family:"Times New Roman"'><o:p></o:p></span></p> <div class=MsoNormal align=center style='text-align:center'><span style='mso-fareast-font-family:"Times New Roman"'> <hr size=5 width="100%" noshade style='color:#C00000' align=center> </span></div> <div class=MsoNormal align=center style='text-align:center'><span style='mso-fareast-font-family:"Times New Roman"'> <hr size=10 width="100%" noshade style='color:black;mso-themecolor:text1' align=center> </span></div> <p align=center style='text-align:center'><span class=style41><span style='font-size:36.0pt'>Exploration 3: Roots of Quadratic Functions</span></span><span style='font-size:36.0pt'><o:p></o:p></span></p> <p align=center style='text-align:center'><span class=style41><span style='font-size:24.0pt'>By</span></span></p> <p align=center style='text-align:center'><span class=style41><span style='font-size:24.0pt'>Elizabeth Gieseking</span></span></p> <div class=MsoNormal align=center style='text-align:center'><span style='mso-fareast-font-family:"Times New Roman"'> <hr size=5 width="100%" noshade style='color:black' align=center> </span></div> <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'> <p class=style5><span style='font-size:20.0pt'>All high school students know the quadratic formula </span><!--[if gte msEquation 12]><m:oMath><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>x</m:r><m:r>=</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>b</m:r><m:r></m:r></span></i><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span 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</v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:149.85pt; height:56.95pt'> <v:imagedata src="Quadratic%20Exploration_files/image001.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=200 height=76 src="Quadratic%20Exploration_files/image002.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><span style='mso-spacerun:yes'></span>and some can even <a href="https://www.youtube.com/watch?v=1oSc-TpQqQI">sing it</a>. They can use it to easily calculate the roots of a quadratic equation, but few know where it came from or what it really means. We will start with a derivation of the quadratic equation and then we will examine the effects of changing each of the coefficients by looking at this equation in the <i style='mso-bidi-font-style: normal'>x-b, <span class=SpellE>x-c</span>,</i> and <i style='mso-bidi-font-style: normal'>x-a</i> planes.<o:p></o:p></span></p> <p class=style5><span style='font-size:20.0pt'>The most common derivation of the quadratic equation is obtained by completing the square.<o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-tab-count:2'> </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:149.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image004.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=199 height=50 src="Quadratic%20Exploration_files/image006.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><span style='mso-tab-count:2'> </span><o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-tab-count:1'> </span>Divide through by <i style='mso-bidi-font-style: normal'>a</i><span style='mso-tab-count:1'> </span></span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>x</m:r><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>c</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:48.75pt'> <v:imagedata src="Quadratic%20Exploration_files/image008.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=65 src="Quadratic%20Exploration_files/image011.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-tab-count:1'> </span>Complete the square<span style='mso-tab-count:1'> </span></span><!--[if gte msEquation 12]><m:oMath><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>x</m:r><m:r>+</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:e></m:d></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:e></m:d><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>c</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>-</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:e></m:d></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:28.5pt;mso-text-raise:-28.5pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:300pt;height:57.75pt'> <v:imagedata src="Quadratic%20Exploration_files/image013.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=400 height=77 src="Quadratic%20Exploration_files/image015.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-spacerun:yes'></span><span style='mso-tab-count:1'> </span>Rearrange and simplify<span style='mso-tab-count:1'> </span></span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:e></m:d></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=</m:r><m:r>-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>c</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>4</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>4</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.5pt;mso-text-raise:-25.5pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:263.25pt;height:54.75pt'> <v:imagedata src="Quadratic%20Exploration_files/image017.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=351 height=73 src="Quadratic%20Exploration_files/image018.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><span style='mso-spacerun:yes'></span><o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-tab-count:1'> </span>Take the square root of both sides<span style='mso-tab-count:1'> </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>x</m:r><m:r>+</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=</m:r></span></i><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>4</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:den></m:f></m:e></m:rad><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>=</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:29.5pt;mso-text-raise:-29.5pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:267.75pt;height:63pt'> <v:imagedata src="Quadratic%20Exploration_files/image020.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=357 height=84 src="Quadratic%20Exploration_files/image022.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><o:p></o:p></span></p> <p class=style5 style='tab-stops:1.0in 5.5in'><span style='font-size:20.0pt'><span style='mso-tab-count:1'> </span>Subtract<span style='mso-tab-count:1'> </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>x</m:r><m:r>=</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>b</m:r><m:r></m:r></span></i><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:129.75pt;height:53.25pt'> <v:imagedata src="Quadratic%20Exploration_files/image024.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=173 height=71 src="Quadratic%20Exploration_files/image026.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><o:p></o:p></span></p> <p class=style5><span style='font-size:20.0pt'>In this exploration, we will examine each of the coefficients in the standard form of the quadratic equation: </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0 </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:153.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image003.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=205 height=50 src="Quadratic%20Exploration_files/image028.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'>and their effects on the roots of the equation.<span class=SpellE>We</span> will start by letting </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>a</m:r><m:r>=</m:r><m:r>c</m:r><m:r>=1</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:84.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image005.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=113 height=50 src="Quadratic%20Exploration_files/image030.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'><span style='mso-spacerun:yes'></span>and we get the <span class=GramE>equation </span></span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;font-family: "Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:137.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image007.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=183 height=50 src="Quadratic%20Exploration_files/image032.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt'>.<span style='mso-spacerun:yes'> </span>Next we will graph this equation on the <i style='mso-bidi-font-style:normal'>x-b</i> plane.<span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=style5 align=center style='text-align:center'><span style='font-size: 20.0pt;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_1" o:spid="_x0000_i1040" type="#_x0000_t75" style='width:594.75pt;height:483pt; visibility:visible;mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image009.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=793 height=644 src="Quadratic%20Exploration_files/image009.png" v:shapes="Picture_x0020_1"><![endif]></span><span style='font-size:20.0pt'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>We inserted several horizontal lines corresponding to different values of b.<span style='mso-spacerun:yes'> </span>The points at which each horizontal line crosses the graph of </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:137.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image007.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=183 height=50 src="Quadratic%20Exploration_files/image032.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>correspond to a solution of the equation for that value of b.<span style='mso-spacerun:yes'> </span>For example when b = 2, the line crosses the graph of the function at one point (x, b) = (-1, 2).<span style='mso-spacerun:yes'> </span>This corresponds to the solution of </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+2</m:r><m:r>x</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image010.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=50 src="Quadratic%20Exploration_files/image054.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>which is </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-1.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:67.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image012.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=90 height=50 src="Quadratic%20Exploration_files/image056.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>When b = 3, the line crosses the graph of </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:137.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image007.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=183 height=50 src="Quadratic%20Exploration_files/image032.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>at two points.<span style='mso-spacerun:yes'> </span>We can use the quadratic formula to determine that these two roots <span class=GramE>are </span></span><!--[if gte msEquation 12]><m:oMath><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>3</m:r></span></i><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>9-4</m:r><m:r>"</m:r><m:r>1</m:r><m:r>"</m:r><m:r>1</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>3</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r></m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>5</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:184.5pt;height:52.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image014.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=246 height=70 src="Quadratic%20Exploration_files/image058.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>When b = 1, the line does not cross the graph of the function.<span style='mso-spacerun:yes'> </span>If we use the quadratic formula, we see that the discriminant </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r><m:r>=1-4</m:r><m:r>"</m:r><m:r>1</m:r><m:r>"</m:r><m:r>1=-3&lt;0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:287.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image016.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=383 height=50 src="Quadratic%20Exploration_files/image060.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>which means there are no real roots.<span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>We can also vary the value of c on this graph.<span style='mso-spacerun:yes'> </span>Below are the curves we obtain when c is set at integer values between 1 and 6.<span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_5" o:spid="_x0000_i1039" type="#_x0000_t75" style='width:597pt; height:566.25pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image042.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=796 height=755 src="Quadratic%20Exploration_files/image042.png" v:shapes="Picture_x0020_5"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>We see that as c increases, the lowest value of |b| which produces real roots also increases.<span style='mso-spacerun:yes'> </span><span style='mso-spacerun:yes'></span>We will again examine the discriminant to see why this is true.<span style='mso-spacerun:yes'> </span>In order to have real roots, </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r><m:r>e"</m:r><m:r>0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:114pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image062.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=152 height=50 src="Quadratic%20Exploration_files/image064.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span><span class=GramE>or </span></span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>e"</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:78.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image066.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=105 height=50 src="Quadratic%20Exploration_files/image069.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>Thus, when we set </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>=1</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:48.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image023.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=65 height=50 src="Quadratic%20Exploration_files/image071.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and increase <i style='mso-bidi-font-style: normal'>c</i>, <i style='mso-bidi-font-style:normal'>b</i> increases in proportion <span class=GramE>with </span></span><!--[if gte msEquation 12]><m:oMath><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>c</m:r></span></i></m:e></m:rad></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:23.25pt;height:39.75pt'> <v:imagedata src="Quadratic%20Exploration_files/image025.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=31 height=53 src="Quadratic%20Exploration_files/image073.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>If we instead chose values <span class=GramE>of </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>c</m:r><m:r>d"0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:47.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image027.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=63 height=50 src="Quadratic%20Exploration_files/image075.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, then </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:21pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image029.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=28 height=50 src="Quadratic%20Exploration_files/image077.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>will always be greater than </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>4</m:r><m:r>ac</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:32.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image031.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=43 height=50 src="Quadratic%20Exploration_files/image079.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and we always have real roots as shown below.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_4" o:spid="_x0000_i1038" type="#_x0000_t75" style='width:638.25pt; height:540.75pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image033.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=851 height=721 src="Quadratic%20Exploration_files/image033.png" v:shapes="Picture_x0020_4"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>In our final graph of <i style='mso-bidi-font-style:normal'>b</i> versus <i style='mso-bidi-font-style:normal'>x</i>, we include values of <i style='mso-bidi-font-style:normal'>c</i> from -5 to 5 and we also add the line </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>x</m:r><m:r>+</m:r><m:r>b</m:r><m:r>=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:98.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image050.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=131 height=50 src="Quadratic%20Exploration_files/image081.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_6" o:spid="_x0000_i1037" type="#_x0000_t75" style='width:689.25pt; height:565.5pt;visibility:visible;mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image052.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=919 height=754 src="Quadratic%20Exploration_files/image052.png" v:shapes="Picture_x0020_6"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>This line goes through four points which we know are single roots of our quadratic equations: (x, b) = (-1, 2) and (1, -2) on the curves corresponding to </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:137.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image007.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=183 height=50 src="Quadratic%20Exploration_files/image032.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and (x, b) = (-2, 4) and (2, -4) on the curves corresponding to </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+4=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:141.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image053.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=189 height=50 src="Quadratic%20Exploration_files/image083.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>If we look at the horizontal <i style='mso-bidi-font-style:normal'>b </i>lines, it appears that the line </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>x</m:r><m:r>+</m:r><m:r>b</m:r><m:r>=0</m:r><m:r> </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:99pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image055.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=132 height=50 src="Quadratic%20Exploration_files/image085.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>is halfway between the two roots for that value of <i style='mso-bidi-font-style:normal'>b</i>.<span style='mso-spacerun:yes'> </span>Why would this be true?<span style='mso-spacerun:yes'> </span>If we solve this equation for <i style='mso-bidi-font-style:normal'>x</i>, we see <span class=GramE>that </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:64.5pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image057.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=86 height=64 src="Quadratic%20Exploration_files/image087.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>If we look back at the quadratic formula, we <span class=GramE>see </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=</m:r><m:r>-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r></m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r><m:r>a</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:159pt;height:53.25pt'> <v:imagedata src="Quadratic%20Exploration_files/image059.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=212 height=71 src="Quadratic%20Exploration_files/image089.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>For the equations we are examining, we are keeping <i style='mso-bidi-font-style: normal'>a</i> constant at 1, so this equation <span class=GramE>becomes </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=</m:r><m:r>-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r></m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:rad><m:radPr><m:degHide m:val="on"/><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt; mso-bidi-font-size:20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-hansi-font-family:"Cambria Math";font-style:italic; mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:radPr><m:deg></m:deg><m:e><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>c</m:r></span></i></m:e></m:rad></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:141.75pt;height:53.25pt'> <v:imagedata src="Quadratic%20Exploration_files/image061.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=189 height=71 src="Quadratic%20Exploration_files/image091.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>The two roots calculated by the quadratic formula are symmetric around </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:69pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image063.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=92 height=64 src="Quadratic%20Exploration_files/image093.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>We will look at this by graphing four quadratic equations with </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>=1 </m:r></span></i><span style='font-size:20.0pt;line-height:150%;font-family: "Cambria Math",serif'><m:r><m:rPr><m:nor/></m:rPr>and</m:r><m:r><i style='mso-bidi-font-style:normal'> </i></m:r><m:r><i style='mso-bidi-font-style: normal'>b</i></m:r><m:r><i style='mso-bidi-font-style:normal'>=6.</i></m:r></span></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:141.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image065.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=189 height=50 src="Quadratic%20Exploration_files/image095.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;line-height:150%'><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;mso-no-proof:yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_7" o:spid="_x0000_i1036" type="#_x0000_t75" style='width:494.25pt;height:457.5pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image067.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=659 height=610 src="Quadratic%20Exploration_files/image067.png" v:shapes="Picture_x0020_7"><![endif]></span></i><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%'><o:p></o:p></span></i></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>When <i style='mso-bidi-font-style:normal'>b</i> = 6, the roots are symmetric <span class=GramE>about </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>=-3</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:117pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image068.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=156 height=64 src="Quadratic%20Exploration_files/image097.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>The equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>y</m:r><m:r>=</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+6</m:r><m:r>x</m:r><m:r>+9</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image070.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=50 src="Quadratic%20Exploration_files/image122.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>has one real root at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-3.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:67.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image072.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=90 height=50 src="Quadratic%20Exploration_files/image123.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>The equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>y</m:r><m:r>=</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+6</m:r><m:r>x</m:r><m:r>+8</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image074.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=50 src="Quadratic%20Exploration_files/image124.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>has two real roots at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-3</m:r><m:r> 1 </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:107.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image076.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=143 height=50 src="Quadratic%20Exploration_files/image125.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>or </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-4</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image078.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image126.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span><span class=GramE>and </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-2</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image080.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image127.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>The equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>y</m:r><m:r>=</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+6</m:r><m:r>x</m:r><m:r>+5</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image082.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=50 src="Quadratic%20Exploration_files/image128.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>has two real roots at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-3 </m:r><m:r>2</m:r><m:r> </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:107.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image084.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=143 height=50 src="Quadratic%20Exploration_files/image129.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>or </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-5</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image086.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image130.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span><span class=GramE>and </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-1</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image088.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image131.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>The equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>y</m:r><m:r>=</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+6</m:r><m:r>x</m:r><m:r>+0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:139.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image090.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=186 height=50 src="Quadratic%20Exploration_files/image132.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>has two real roots at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-3 </m:r><m:r>3</m:r><m:r> </m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:107.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image092.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=143 height=50 src="Quadratic%20Exploration_files/image133.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>or </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-6</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image094.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image134.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span><span class=GramE>and </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:62.8pt;height:37.65pt'> <v:imagedata src="Quadratic%20Exploration_files/image096.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image135.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'>In summary, we have found that a quadratic equation of the form </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:136.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image120.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=182 height=50 src="Quadratic%20Exploration_files/image136.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>can have zero, one, or two real roots, depending on the value of c.<span style='mso-spacerun:yes'> </span>If </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r><m:r>&gt;0</m:r><m:r>,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:117.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image137.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=157 height=50 src="Quadratic%20Exploration_files/image138.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>then there are two real roots symmetric about </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:69pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image063.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=92 height=64 src="Quadratic%20Exploration_files/image093.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span><span class=GramE>If </span></span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r><m:r>=</m:r><m:r>0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:114pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image139.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=152 height=50 src="Quadratic%20Exploration_files/image140.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, then there is one root of multiplicity 2 at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:69pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image063.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=92 height=64 src="Quadratic%20Exploration_files/image093.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>If </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r><m:r>4</m:r><m:r>ac</m:r><m:r>&lt;</m:r><m:r>0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:114pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image141.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=152 height=50 src="Quadratic%20Exploration_files/image142.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, then there are no real roots.<span style='mso-spacerun:yes'> </span>There are however two complex roots symmetric about </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.0pt;mso-text-raise:-25.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:69pt;height:48pt'> <v:imagedata src="Quadratic%20Exploration_files/image063.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=92 height=64 src="Quadratic%20Exploration_files/image093.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>Next we will examine the equation </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:136.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image120.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=182 height=50 src="Quadratic%20Exploration_files/image136.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>in the <span class=SpellE><i style='mso-bidi-font-style: normal'>x-c</i></span> plane.<span style='mso-spacerun:yes'> </span>In the following diagram, b = 4.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_9" o:spid="_x0000_i1035" type="#_x0000_t75" style='width:525.75pt;height:491.25pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image143.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=701 height=655 src="Quadratic%20Exploration_files/image143.png" v:shapes="Picture_x0020_9"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>The horizontal lines correspond to different values of c.<span style='mso-spacerun:yes'> </span>The points at which the graph crosses each value of c are the roots of </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+4</m:r><m:r>x</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:141pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image144.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=188 height=50 src="Quadratic%20Exploration_files/image145.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>For example, in this graph we see that the roots of </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+4</m:r><m:r>x</m:r><m:r>+3=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:138pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image146.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=184 height=50 src="Quadratic%20Exploration_files/image147.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>are </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-3</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image148.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image149.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-1</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image088.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=84 height=50 src="Quadratic%20Exploration_files/image131.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+4</m:r><m:r>x</m:r><m:r>+4</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:100.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image150.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=134 height=50 src="Quadratic%20Exploration_files/image151.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>has one root at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-2,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:67.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image152.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=90 height=50 src="Quadratic%20Exploration_files/image153.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and if </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>c</m:r><m:r>&gt;4,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:51pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image154.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=68 height=50 src="Quadratic%20Exploration_files/image155.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>there are no real roots.<span style='mso-spacerun:yes'> </span>We can graph the equation with multiple values of b on the same set of <span class=SpellE><i style='mso-bidi-font-style: normal'>x-c</i></span> axes.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_13" o:spid="_x0000_i1034" type="#_x0000_t75" style='width:861.75pt;height:545.25pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image174.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=1149 height=727 src="Quadratic%20Exploration_files/image174.png" v:shapes="Picture_x0020_13"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>Here we have a series of downward facing parabolas.<span style='mso-spacerun:yes'> </span>As the value of <i style='mso-bidi-font-style:normal'>b</i> increases from 0, the vertex of the parabola moves upward and to the left.<span style='mso-spacerun:yes'> </span>As it decreases from 0, the vertex moves upward and to the right.<span style='mso-spacerun:yes'> </span>If we look back at the equation we see that the vertex is <span class=GramE>at </span></span><!--[if gte msEquation 12]><m:oMath><m:d><m:dPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>-</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:den></m:f><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>,</m:r></span></i><m:f><m:fPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size: 20.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>b</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup></m:num><m:den><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt; line-height:150%;font-family:"Cambria Math",serif'><m:r>4</m:r></span></i></m:den></m:f></m:e></m:d></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:25.5pt;mso-text-raise:-25.5pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:74.25pt;height:51.75pt'> <v:imagedata src="Quadratic%20Exploration_files/image175.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=99 height=69 src="Quadratic%20Exploration_files/image156.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>.<span style='mso-spacerun:yes'> </span>We also note that all of these parabolas pass through (x, c) = (0, 0).<span style='mso-spacerun:yes'> </span>This is due to the fact that when c = 0, the quadratic can be factored as </span><!--[if gte msEquation 12]><m:oMath><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math"'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>x</m:r></span></m:e><m:sup><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r><m:rPr><m:scr m:val="roman"/><m:sty m:val="p"/></m:rPr>2</m:r></span></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>=</m:r><m:r>x</m:r><m:r>(</m:r><m:r>x</m:r><m:r>+</m:r><m:r>b</m:r><m:r>)</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:165.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image157.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=221 height=50 src="Quadratic%20Exploration_files/image158.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>which has a root at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:48.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image178.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=65 height=50 src="Quadratic%20Exploration_files/image159.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and a second root <span class=GramE>at </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=-</m:r><m:r>b</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:63.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image180.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=85 height=50 src="Quadratic%20Exploration_files/image160.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%'>.<o:p></o:p></span></i></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>Our final exploration will take place on the <i style='mso-bidi-font-style:normal'>x-a</i> plane.<span style='mso-spacerun:yes'> </span>We will start with the equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+1=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:150.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image182.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=201 height=50 src="Quadratic%20Exploration_files/image183.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>and vary <i style='mso-bidi-font-style:normal'>b</i> from 0 to 5.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_16" o:spid="_x0000_i1033" type="#_x0000_t75" style='width:655.5pt;height:452.25pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image184.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=874 height=603 src="Quadratic%20Exploration_files/image184.png" v:shapes="Picture_x0020_16"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>The first thing we notice is that there is an asymptote at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image204.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image186.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>When </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=0,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image206.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image188.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>there is no value for the coefficient <span class=GramE><i style='mso-bidi-font-style:normal'>a </i><span style='mso-spacerun:yes'></span>which</span> produces a solution of this equation.<span style='mso-spacerun:yes'> </span>We see that when </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r><m:r>&gt;0,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image185.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image190.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>if </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:48.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image187.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=65 height=50 src="Quadratic%20Exploration_files/image192.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>there is one root, corresponding to the linear equation </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>bx</m:r><m:r>+1=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:98.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image189.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=131 height=50 src="Quadratic%20Exploration_files/image194.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span><span class=GramE>If</span> </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>b</m:r><m:r>=0,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image191.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image196.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>there is no root at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>=0,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image193.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image198.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>since that would imply that </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>1=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image195.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image200.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span>When </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>&gt;0,</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:53.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image197.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=71 height=50 src="Quadratic%20Exploration_files/image219.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>there may be zero, one, or two real roots, all at negative values of <i style='mso-bidi-font-style:normal'>x.</i><span style='mso-spacerun:yes'> </span><span class=GramE>When </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r><m:r>&lt;0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:48.75pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image199.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=65 height=50 src="Quadratic%20Exploration_files/image220.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, there are always two real roots, one positive and one negative.<span style='mso-spacerun:yes'> </span>If we consider negative values of <i style='mso-bidi-font-style:normal'>b</i>, our graph is the reflection of this graph over the <i style='mso-bidi-font-style: normal'>a</i>-axis.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_17" o:spid="_x0000_i1032" type="#_x0000_t75" style='width:667.5pt;height:494.25pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image201.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=890 height=659 src="Quadratic%20Exploration_files/image201.png" v:shapes="Picture_x0020_17"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>We will also consider the graph of </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>a</m:r></span></i><m:sSup><m:sSupPr><span style='font-size:20.0pt;mso-ansi-font-size:20.0pt;mso-bidi-font-size:20.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><i style='mso-bidi-font-style: normal'><span style='font-size:20.0pt;line-height:150%;font-family:"Cambria Math",serif'><m:r>x</m:r></span></i></m:e><m:sup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>2</m:r></span></i></m:sup></m:sSup><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>+</m:r><m:r>bx</m:r><m:r>+</m:r><m:r>c</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:149.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image004.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=199 height=50 src="Quadratic%20Exploration_files/image006.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'></span>when we keep <i style='mso-bidi-font-style: normal'>b</i> constant at -4 and vary <i style='mso-bidi-font-style:normal'>c</i> from 0 to 5.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_18" o:spid="_x0000_i1031" type="#_x0000_t75" style='width:725.25pt;height:518.25pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image202.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=967 height=691 src="Quadratic%20Exploration_files/image202.png" v:shapes="Picture_x0020_18"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='text-indent:.5in;line-height:150%'><span style='font-size:20.0pt;line-height:150%'>We still have an asymptote at </span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>x</m:r><m:r>=0.</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:52.5pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image204.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=70 height=50 src="Quadratic%20Exploration_files/image186.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'><span style='mso-spacerun:yes'> </span><span class=GramE>When </span></span><!--[if gte msEquation 12]><m:oMath><i style='mso-bidi-font-style:normal'><span style='font-size:20.0pt;line-height: 150%;font-family:"Cambria Math",serif'><m:r>c</m:r><m:r>=0</m:r></span></i></m:oMath><![endif]--><![if !msEquation]><span style='font-size:11.0pt;line-height:107%;font-family:"Calibri",sans-serif; mso-ascii-theme-font:minor-latin;mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast;mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman";mso-bidi-theme-font:minor-bidi; position:relative;top:19.0pt;mso-text-raise:-19.0pt;mso-ansi-language:EN-US; mso-fareast-language:EN-US;mso-bidi-language:AR-SA'><!--[if gte vml 1]><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:47.25pt;height:37.5pt'> <v:imagedata src="Quadratic%20Exploration_files/image216.png" o:title="" chromakey="white"/> </v:shape><![endif]--><![if !vml]><img width=63 height=50 src="Quadratic%20Exploration_files/image221.png" v:shapes="_x0000_i1025"><![endif]></span><![endif]><span style='font-size:20.0pt;line-height:150%'>, <i style='mso-bidi-font-style:normal'>a</i> tends toward negative infinity when <i style='mso-bidi-font-style:normal'>x</i> approaches zero from the negative side and positive infinity when <i style='mso-bidi-font-style:normal'>x</i> approaches zero from the positive side.<span style='mso-spacerun:yes'> </span>For all other positive values of <i style='mso-bidi-font-style:normal'>c</i>, <span class=GramE><i style='mso-bidi-font-style:normal'>a</i> tends</span> toward negative infinity when <i style='mso-bidi-font-style:normal'>x</i> approaches zero from either direction.<span style='mso-spacerun:yes'> </span>When we choose negative values for <i style='mso-bidi-font-style:normal'>c</i>, the graph is reflected over the origin.<o:p></o:p></span></p> <p class=style5 align=center style='text-align:center;text-indent:.5in; line-height:150%'><span style='font-size:20.0pt;line-height:150%;mso-no-proof: yes'><!--[if gte vml 1]><v:shape id="Picture_x0020_19" o:spid="_x0000_i1030" type="#_x0000_t75" style='width:651.75pt;height:561.75pt;visibility:visible; mso-wrap-style:square'> <v:imagedata src="Quadratic%20Exploration_files/image218.png" o:title=""/> </v:shape><![endif]--><![if !vml]><img border=0 width=869 height=749 src="Quadratic%20Exploration_files/image218.png" v:shapes="Picture_x0020_19"><![endif]></span><span style='font-size:20.0pt;line-height:150%'><o:p></o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'><o:p>&nbsp;</o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'><o:p>&nbsp;</o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'><o:p>&nbsp;</o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'><o:p>&nbsp;</o:p></span></p> <p class=style5 style='line-height:150%'><span style='font-size:20.0pt; line-height:150%'><o:p>&nbsp;</o:p> </span><a href="../Elizabeth.html">Return to Elizabeth's Home Page</a></p> </blockquote> <p class=style5 align=center style='text-align:center'>&nbsp;</p> <blockquote style='margin-top:5.0pt;margin-bottom:5.0pt'> <p class=MsoNormal><span style='mso-fareast-font-family:"Times New Roman"'>&nbsp;<o:p></o:p></span></p> </blockquote> </div> </body> </html>