Assignment 3:

Exploring the Graphs of the Equations x2 + 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 x2 + bx + c = 0 in the plane xb for various values of c and the number and types of roots of such equations.

Consider the graph of the equation x2 + bx + 1 = 0 in the plane xb.

If we overlay the graph of b = constant to the graph above, their intersections are the roots of the equation x2 + bx + 1 = 0.

Example for b = -4

Note 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 x2 + bx + c = 0 in the plane xb for various values of c.

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 of b.

Case 2. c = 0

When c = 0, according to this graph the equation has only one real root for any value of b, specifically the solution x = -b.

But if we look to the equation x2 + bx + c = 0 with c = 0,

x2 + bx = 0

x (x + b) = 0

x = 0 and x + b = 0

x = 0 and x = -b

Now, 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.

x2 + bx = 0

bx = -x2

b = -x2x where x ≠ 0

b = -x

Therefore for c = 0, we always have two real roots x = 0 and x = -b.

Case 3. c > 0

Here for different values of b we have 0, 1, or 2 real roots for the equation.

Now lets explore for which values of b we 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 x2 + bx + c = 0 for c > 0 in the xb plane have slope 0.

Differentiating.

Equating to zero.

Evaluating the equation in this values of x we get:

So when c > 0 the equation x2 + 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 b2 - 4ac, which gave us information about the number and type of solutions of a quadratic equation.