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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