Assignment 3

Consider the equation

Now graph this relation in the xb plane. We get the following graph.

If we take any particular value of b, say b = 5, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of the original equation for that value of b. So when b=5, there are two negative real roots, seen on the following graph.

For each value of b we select, we get a horizontal line. Let us generalize about these roots:

b > 2

two negative real roots

b = 2 one negative real root
-2 < b < 2 no real roots
b = -2 one positive real root

b < -2

two positive real roots

Why is this?

Well, a quadratic equation has solutions called roots. These two solutions may or may not be distinct and they may of may not be real.

Let's take a look at the quadratic formula...

the expression underneath the square root sign is called the discriminant.

If the discriminant is positive, there are two distinct real roots. If the discriminant is zero, there is exactly one distinct real root. If the discriminant in negative, there are no real roots.

Giving us both of the negative real roots on the above graph.

Consider the case when c = - 1 rather than + 1.

There is one negative real root and one positive real root.

What would c have to be, for b=5 to give one root.

The discriminant would have to equal zero.

This gives one negative root. The positive root would be given with the line b = -5.

We want to see what happens when c varies, so we call it n. b is refered to as y.

As n varies, what line gives us one root?

The following animation shows this for positive and negative y, as well as positive and negative n.

Graph

n<0

n>0

Graph

n<0

n>0