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The Discriminant

In the quadratic formula, the expression under the square root sign, b² – 4ac, is called the discriminant.

The sign of the discriminant can be used to find the number of solutions of the corresponding quadratic equation,

ax² + bx + c = 0

If the discriminant b² – 4ac is negative, then there are no real solutions of the equation . (You need complex numbers to deal with this case properly. These are usually taught in Algebra 2.)

If the discriminant is zero, there is only one solution.

If the discriminant is positive, then the ± symbol means you get two answers.

The solutions of this equation correspond to the x-intercepts of the parabola

y = ax² + bx + c.

So, you can also use the discriminant to find the number of x-intercepts of a parabola.

Parabola with two x-intercepts (positive discriminant)

 

Parabola with one x-intercept (zero discriminant)

 

Parabola with no x-intercept (negative discriminant)

 

Here a = 1, b = –1, and c = –12. Substituting, we get:

Simplify.

The discriminant is positive, so we have two solutions:

x = 4 and x = –3

In this example, the discriminant was 49, a perfect square, so we ended up with rational answers. Often, when using the quadratic formula, you end up with answers which still contain radicals.

Here a = 3, b = 2, and c = 1. Substituting, we get:

Simplify.

The discriminant is negative, so this equation has no real solutions.