Finding Quadratic Roots

Not using algebraic methods, that is.

By: Carly Cantrell

In the xb plane:

It is determined that changing different parameters will affect the graph of quadratics in standard form and in vertex form.

This exploration explores the same quadratic equations, but now in the xb plane.

Let’s consider the following equation’s graph:

x²+ bx +1 = 0

Note: the parameter c is 1.

What about when c is varying?

The following graphs show the varying values of c, when c 1.

The following graphs show the varying values of c, when c 0.

Notice: when c = 0, there is a line at x = -b

Let’s look at all the varying values on the same graph; we have a collection of hyperbolas:

Now, let’s get to the good stuff. When graphing quadratics in the xb plane, we are able to plot horizontal lines.

You plot these in the form of b=c where c is a constant and then the intersecting points are the roots.

The number of intersections entails how many roots that quadratic will have.

The pattern!

Our pattern is supported by the quadratic formula.

So, when 2, there are no real solutions.

When 2 there is exactly one real solution.

When 2 there are two real solutions.