3. The Quadratic Formula

At the end of the last section (Completing the Square), we derived a general formula for solving quadratic equations. Here is that general formula:

For any quadratic equation `ax^2+ bx + c = 0`, the solutions for x can be found by using the quadratic formula:


The expression under the square root, `b^2− 4ac`, can tell us how many roots we'll get. (There's no magic here - just a consideration of what the square root of `b^2− 4ac` is.)

If ` b^2− 4ac = 0`, then we'll have one root only, `x = −b/(2a)`.

If ` b^2− 4ac > 0`, then we'll have two roots, one involving the "+" sign and the other involving the "−" sign in the formula.

If ` b^2− 4ac < 0`, then we'll have no real roots, since you cannot find the square root of a negative number.

The expression `b^2 − 4ac` is called the discriminant and in some books you will see it written with a Greek upper case Delta, like this `Delta = b^2 − 4ac`.

Example 1

Solve `2x^2- 7x - 5 = 0` using the quadratic formula.

Example 2

Solve `2x^2= 4x + 3`


Solve `6r^2= 6r + 1` using the quadratic formula.