Presented by

Dana TeCroney

The As, Bs,
and Cs of Quadratics

The purpose of this investigation is to explore what happens as
the values of the coefficient ÔbÕ varies in a quadratic equation.

Consider the following:

y = ax^{2} +bx + c

If
we let a = 1 and c = 1, what would the nature of this equation be? Consider the roots, what would they be
(and how many)?

To
explore this, consider the equation

0 = x^{2} + bx + 1

In
the xy plane the value of b will be one factor that determines the number of
roots this equation has. For an
example press HERE.

An
interesting way to look at this would be in the xb plane.

Another
way of thinking about the number of roots is to draw in some horizontal lines
corresponding to different b values:

As
you can see by the graph, when b > 2 or when b < -2 there are two roots
to the equation since the line intersect the curve twice. When –2 < b < 2 there are no
real roots since there are no intersection points. This implies that at b = -2, or b = 2 there is one
intersection, or one root. Can we
see this another way?

Given
the equation x^{2} + bx +1 = 0, the roots can be found by

When
will this have one root?
When the discriminate equals zero of course, ie. when b = -2 or when b =
2.

Another
interesting problem would be to change the value of a or c, would this change
the nature of the answers? First
letÕs look at a = -1. So, -x^{2}
+ bx + 1 = 0, or .
In the xb
plane, this would look like:

So,
if any horizontal line were to be drawn in what would happen, how many
intersections would there be? The
answer of course is two, and so there will always be two roots. To confirm this you could also look at
the discriminate in the quadratic equation:

The
discriminate of which will never be zero (for real b).

Now
letÕs consider changing ÔcÕ. Let a
= 1 again and let c = -1.

x^{2} + bx - 1 = 0,
or

In
the xb plane, this would look like:

Again
by the same arguments as above this equation will have two roots for c = -1,
and in fact for c < 0.