The quadratic formula, first discovered by the Babylonians four thousand years ago, gives you a surefire way to solve quadratic equations of the form

0 = ax2 + bx + c.

Plugging in the values of a, b, and c, you will get the desired values of x.

If the expression under the square root sign (b2 – 4ac, also called the discriminant) is negative, then there are no real solutions. (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.

Example 1:

x2x – 12 = 0

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.

Example 2:

3x2 + 2x + 1 = 0

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

Simplify.

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

Example 3: