### The Discriminant of a Quadratic Equation Part 1

Recall that the quadratic equation ax2bx + c = 0 can be solved using the quadratic formula:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Before we go on, you may also want to review the information on square roots that can be found here:

Square Root Basics

The discriminant of the quadratic equation ax2bx + c = 0 is the quantity Δ defined by

Δ = b2 – 4ac

In other words, the discriminant is simply the expression that appears under the square root in the quadratic formula.

Although computing the discriminant of a quadratic equation does not give the roots (solutions) of the equation, it does give us a lot of information about the nature of the roots and the graph of the equation.

For example, if the discriminant of the quadratic equation ax2bx + c = 0 is 0, then the quadratic formula simplifies to

x = –b/2a,

and we see that there is just one solution.

If the coefficients a and b are integers, then the unique solution will be a rational number.

Graphically, this means that the graph of the function y = ax2 + bx + c is a parabola that intersects the x-axis at one point.

An example is given by the red parabola in the image above.

Note: The discriminant does not tell us if the parabola opens upwards or downwards. However, this is easy to determine simply by looking at the value of a.

If a > 0 (i.e. a is a positive number), then the parabola opens upwards.

If a < 0 (i.e. a is a negative number), then the parabola opens downwards.

So, the red parabola in the image above has an equation of the form y = ax2 + bx + c where < 0 and Δ = 0

Example: Find the discriminant of x2 + 6+ 9 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x2 + 6x + 9.

I will post a solution to this problem tomorrow, and then discuss other possibilities for the determinant. Feel free to post your own solutions in the comments.