# 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:

`x=(-b+-sqrt(b^2-4ac))/(2a)`

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`

### Exercise

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

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