**Assignment 3:**

** Exploring the Graphs of the Equations x^{2} + bx + c = 0 in the Plane xb for Various Values of c**

by

Ángel M. Carreras Jusino

Goal:

- Find a relationship between the graphs of the equations of the form
x^{2}+bx+c= 0 in the planexbfor various values ofcand the number and types of roots of such equations.

Consider the graph of the equation

x^{2}+bx+ 1 = 0 in the planexb.If we overlay the graph of

b= constant to the graph above, their intersections are the roots of the equationx^{2}+bx+ 1 = 0.Example for

b= -4Note that for

|b|= 2 there is one real solution, |b| > 2 there are two real solutions, and for |b| < 2 there is no real solution. See the following animation.

Now lets explore the graphs of the equations of the form

x^{2}+bx+c= 0 in the planexbfor various values ofc.

From the animation above can be noticed that:

Let explore each case.

Case 1.

c< 0

When

c< 0, we always have two real roots no matter what is the value ofb.Case 2.

c= 0

When

c= 0, according to this graph the equation has only one real root for any value ofb, specifically the solutionx=-b.But if we look to the equation

x^{2}+bx+c= 0 withc= 0,

x^{2}+bx= 0

x(x +b) = 0

x= 0 andx+b= 0

x= 0 andx= -bNow, why the solution

x= 0 doesn't appear in the graph?This happen because what is graph in the graphing software was an algebraic manipulation of the original equation which exclude 0 from its domain.

x^{2}+bx= 0

bx= -x^{2}

b=^{-x2}⁄_{x}wherex≠ 0

b= -xTherefore for

c= 0, we always have two real rootsx= 0 andx= -b.Case 3.

c> 0

Here for different values of

bwe have 0, 1, or 2 real roots for the equation.Now lets explore for which values of

bwe have the different number of real roots.Looking at the animation we can note that the equation have one real root when the graph of the equations

x^{2}+bx+c= 0 forc> 0 in thexbplane have slope 0.Differentiating.

Equating to zero.

Evaluating the equation in this values of

xwe get:

So when

c> 0 the equationx^{2}+bx+c= 0 has:no real solution if

b< |2√c|one real solution if

b= |2√c|two real solution if

b> |2√c|Note that this is equivalent to the discriminant

b^{2}- 4ac, which gave us information about the number and type of solutions of a quadratic equation.