University of Georgia

Assignment #3 for EMAT 6680

In one of the high schools where the second author used to teach, if you asked students to solve a quadratic equation,

the FIRST thing many would do is unsheath their graphing calculators and tell you that they were going to "graph it" and find the x-intercepts. (Some would tell you they were going to "graph the parabola" and graph the x-axis and then find the intersection points.) This now rather standard technique was certainly not prominent when either author was in high school! And of course, it has some limitations, since complex solutions are not readily obtainable. It also can obscure the difference between an equation and a function.

Therefore, the aim of this paper is to go beyond the "graph and find zeros" or "graph and intersect" techniques and look at ways to characterize the roots of a quadratic equation by examining its related quadratic function and other relations. That is, we will graph the quadratic function

for different values of **a**,
**b**, or **c**
as the other two values are held constant. From these graphs,
we can discuss patterns of roots for the corresponding quadratic
equation, **ax^2 + bx
+ c = 0** by examining the locus
of vertices as **a**, **b**, or **c**
varies. However, we can actually go a step farther and consider
the x**a**, x**b**,
and x**c** planes, and how relations
graphed there can give us further insight into the nature of the
roots of the quadratic equation.

To navigate through the paper, you can choose to jump to discussions of

Consider the graphs below of **y =** **ax^2
+ x + 1** with the values of **a**
varying as indicated. Obviously, for some values of **a**,
there will be no real solutions for the corresponding quadratic
equation, **ax^2 + x + 1 = 0**.
In particular, for **a** greater
than about 0.25, the quadratic function appears to have no real
x-intercepts or zeros, and so the corresponding quadratic equation
would have no real roots.

To determine the exact value of **a**
for which the quadratic equation will have exactly one real root,
we have to try to find the value of **a**
that "gets the parabola tangent" to the x-axis. *Note
that though it is common to call that one real root a double root,
for sake of clarity throughout this paper, we will refer to a
double root as "one real root."*

Examining the locus of vertices is both interesting and helpful
in our quest to find this key value of **a**.
As you can see below, the locus of vertices forms a line with
equation **y = 0.5x + 1**.

For more information about the locus of vertices
as a varies, see this proof and dicussion
of the significance of the function that describes the locus. |

Notice that to find the value of **x** for which **ax^2 + x + 1 = 0** has one real root,
we simply want to find the zero of the "blue graph,"
**y = 0.5x + 1**.
Algebra tells us the zero is x = -2, which we can confirm visually
with the graph. Therefore, when x = -2, the quadratic equation
**ax^2 + x + 1 = 0 **will have one real root. To find the key value
of **a**
for which the equation has one real root, we can substitute
x = -2 into the equation and solve for **a**.
We get **a** = 0.25, as we suspected.

Note also that at **a** = 0,
the "quadratic" becomes a line with equation **y =
x + 1** (with a single zero at x = -1.) This observation is
consistent with the conjecture that for values of **a**
below 0.25 the corresponding quadratic equation will have at least
one real root; however, presumably for values of **a**
between 0 and 0.25, there will be two roots, as there certainly
are two real roots for values of **a**
less than 0.

Here's one way to visualize how the value of **a**
affects the zeros of the quadratic function and therefore the
types of roots of the corresponding quadratic equation. Consider
the graph of **yx^2 + x + 1 = 0**, where we think of the y-axis
as the **a**-axis.

The red graph is yx^2 + x + 1 = 0 (or
ax^2 + x + 1 = 0), and the blue line is y = -1, or a = -1. |
The intersections of the relation and line occur at x = 1.62 and x = -.62. Notice that the roots of So, for specific values of |

As you can see, the maximum value occurs when the value of
**a** is 0.25. For this value
of **a**, the one real root (or
double root) of **0.25x^2 + x +
1 = 0** will be x = -2.

To summarize, for all values of **a**
> 0.25, the quadratic equation **ax^2
+ x + 1 = 0** has no real solutions; for **a**
= 0.25, there is one real root; and for **a**
< 0.25, there are two real solutions except at **a**
= 0, in which case the quadratic equation "degenerates"
into a linear equation with one root at x = -1.

Of course, other values of **b**
and **c** will affect the graphs
for varying values of **a**, both
in the xy and x**a** planes. Below
are two related graphs when **b**
= -3 and **c** = -2. The first
graph shows **y =** **ax^2 -
3x - 2** for varying values of **a**.
The locus of vertices, **y = -1.5x - 2**,
is also graphed in blue. You can see that this time for values
of **a** lower than a little less
than -1, the graph ceases to have real roots.

In fact, at x = -4/3, (the zero of **y
= -1.5x - 2**), the quadratic **ax^2
- 3x - 2 = 0** should have exactly one real root. Therefore,
(subsituting x = -4/3 and solving for **a**),
we find that at **a** = -1.125,
the quadratic will have exactly one real root.

And in fact, as shown by the graph of the x**a**
plane (i.e. the graph of **yx^2 - 3x -2 = 0** or **ax^2
-3x - 2 = 0**), the minimum value occurs at about **a**
= -1.12. For that value of **a**,
the quadratic equation will have one real (or double) root of
about -1.3. For **a** greater
than approximately -1.12, the equation will have two real roots,
and for **a** less than approximately
-1.12, the equation will have no real roots.

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Now consider the graphs below
of **y = x^2 + bx + 1** with the
values of **b** varying as indicated.
Obviously, for some values of **b**,
there will be no real roots for the corresponding quadratic equation,
**x^2 + bx + 1 = 0**. In particular,
for -2 < **b** < 2, or |**b**| < 2, the quadratic function
appears to have no real x-intercepts or zeros, and so the corresponding
quadratic equation has no real roots.

Unlike our investigation with **a**,
there appear to be two values of **b**
for which the quadratic equation will have exactly one real root.
Also notice that when **b** =
0, the resulting function and corresponding equation are still
quadratic, i.e., **y = x^2 + 1**, unlike when **a**
= 0. Similar to our investigation of **a**,
however, we are still interested in the values of **b**
that "get the parabola tangent" to the x-axis.

Examining the locus of vertices of the parabolas as **b** varies is also instructive in this
situation. This time the locus of vertices forms a parabola, whose
equation is **y = -x^2 + 1**,
as you can see by viewing the blue graph below.

For more information about the locus of vertices
as b varies, see this proof and discussion
of the significance of the function that describes the locus. |

Since **y = -x^2 + 1 **has
positive y-values for |x| < 1, the vertices for the purple
parabolas will lie above the x-axis for |x| < 1. Since **a** is greater than 0 in the function
**y = x^2 + bx + 1**, all parabolas
"open up," and so we can deduce that for |x| < 1,
the quadratic equation **x^2 + bx
+ 1 = 0** has no real roots. That is, for |x| < 1, which
means for |**b**| < 2, these
quadratic functions cannot intersect the x-axis. Similarly, since
the **y = -x^2 + 1** has negative
y-values for |x| > 1, the quadratic equation **x^2 + bx + 1 = 0** will have two real roots
for |x| > 1, i.e., for |**b**|
> 2.

Examining the x**b** plane
can provide further some further insight and confirmation. Consider
the graph of **x^2 + yx + 1 = 0**, where we think of the y-axis
as the **b**-axis.

The purple graph is x^2 + yx + 1 = 0 (or
x^2 + bx + 1 = 0), the black line is y = - 4 or b = - 4;and the blue lines are y = - 2 and y
= 2, or b
= - 2 and b = 2. |
As with the x For Similarly, for Finally, it is easy to see that for |

Here's another example, with both the xy plane and x**b** plane shown for **a**
= -1.5 and **c** = 2. That is,
the first graph shows **y = -1.5x^2 + bx
+ 2** for varying values of **b**
(integers from -4 to 4). Notice that the locus of vertices, **y = 1.5x^2 + 2** (shown in blue), does
not intersect the x-axis. Therefore, the purple graphs will *never*
become tangent to the x-axis! So there will never be only a single
real root for the corresponding quadratic equation. And in fact,
since the value of **a** <
0, the purple parabolas will always "open down" and
intersect the x-axis in 2 places. So we claim that the corresponding
quadratic equation will *always* have two real roots.

And in fact, as shown by the graph of of the x**b**
plane (i.e. the graph of **-1.5x^2 + yx + 2 = 0** or **-1.5x^2 + bx
+ 2 = 0**), it is clear that every horizontal line will intersect
the hyperbola in two places. So our claim is justified!

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Fortunately, analyzing the nature of the roots of the quadratic
equation **x^2 + x + c = 0** as
**c** varies is relatively simpler
than the analysis for **a** and
**b**! Many people know even without
graphing that changing **c** merely
translates the graph vertically, as shown below. Thus, the locus
of vertices is a vertical line (shown in blue), **x
= -0.5**.

For more information about the locus of vertices
as c varies, see this proof and discussion
of the significance of the function that describes the locus. |

In these graphs of **y =** **x^2 + x + c**,
**c** varies through the integers
from -4 to 4, and we can clearly see that the function will have
a single real zero for a value of **c**
between 0 and 1. For **c**-values
greater than this key value, there will be no real roots of the
corresponding quadratic equation, and for **c**-values
less than this key value, there will be two real roots.

Again, the intersection of the locus and the x-axis tells us
the x-coordinate (x = -0.5) for which the parabola **y =**
**x^2 + x + c **will be tangent
to the x-axis. Subsituting this value of x into **x^2 + x + c = 0** and solving for **c**,
we find that **c** = 0.25, the
key value that allows us to characterize the roots of the equation.

Again, to confirm our analysis, let's examine the graph of
**x^2 + x + y = 0**,
or **x^2 + x + c
= 0, **where we think of the y-axis as the **c**-axis.
It shouldn't be too much of a surprise that we produce a parabola,
**y = -x^2 - x**, or **c = -x^2
- x**. The vertex of this parabola give us the **c**-value
for which there is one real root to the quadratic equation; namely,
at **c** = 0.25, the single real
root to **x^2 + x + 0.25
= 0** is x = -0.5.

Furthermore, the blue line, which represents
**y = -2** or **c = -2**, intersects
the graph at x =1 and x = -2, which are indeed the roots of **x^2
+ x + -2 =
0**.

So, in summary, for **c** >
0.25, there will be no real roots to the quadratic equation **x^2
+ x + c = 0**; for **c** = 0.25 there will be a single real root; and
for **c**
< 0.25 there will be two real roots.

We hope you have found it instructive
(if not exhaustive!) to examine the loci of vertices and to view
the xy and x**a**, x**b**, and x**c** planes in examining the nature of the roots
of standard quadratic equations. Certainly the mulitple views
can provide further insight for both teachers and students into
the analysis of a standard class of functions and equations in
the high school curriculum.

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