# Solve using the quadratic formula

This lesson shows how to solve using the quadratic formula. To use the quadratic formula, you need to identify a, b, and c using the standard form of a quadratic equation.  You can also write the quadratic formula as x = [-b ± √(b2 - 4ac)]/2a

## How to identify a, b, and c

The standard form is ax2 + bx + c = 0

1) for 6x2 + 8x + 7 = 0, we get a = 6, b = 8, and c = 7

2) for x2 + 8x - 7 = 0, we get a = 1, b = 8, and c = -7

2) for -x2 - 8x + 7 = 0, we get a = -1, b = -8, and c = 7

## More examples showing how to solve using the quadratic formula

Example #1:

Solve using the quadratic formula x2 + 8x + 7 = 0

a = 1, b = 8, and c = 7

x = [-b ± √(b2 - 4ac))] / 2a

x = [-8 ± √(82 - 4 × 1 × 7)] / 2 × 1

x = [-8 ± √(64 - 4 × 1 × 7)] / 2

x = [-8 ± √(64 - 4 × 7)] / 2

x = [-8 ± √(64 - 28)] / 2

x = [-8 ± √(36)] / 2

x = (-8 ± 6 ) / 2

x1 = (-8 + 6 ) / 2

x1 = (-2 ) / 2

x1 = -1

x2 = (-8 - 6 ) / 2

x2 = (-14 ) / 2

x2 = -7

Example #2:

Solve using the quadratic formula 4x2 - 11x - 3 = 0

a = 4, b = -11, and c = -3

x = [-b ± √(b2 - 4ac)] / 2a

x = [- -11 ± √( (-11)2  - 4 × 4 × -3)] / 2 × 4

x = [11 ± √(121 - 4 × 4 × -3)] / 8

x = [11 ± √(121 - 4 × -12)] / 8

x = [11 ± √(121 + 48)] / 8

x = [11 ± √(169)] / 8

x = (11 ± 13 ) / 8

x1 = (11 + 13 ) / 8

x1 = (24 ) / 8

x1 = 3

x2 = (11 - 13 ) / 8

x2 = (-2 ) / 8

x2 = -1/4

Example #3:

Solve using the quadratic formula x2 + x - 2 = 0

a = 1, b = 1, and c = -2

x = [-b ± √(b2 - 4ac)] / 2a

x = [- 1 ± √( (1)2  - 4 × 1 × -2)] / 2 × 1

x = [-1 ± √(1 - 4 × 1 × -2)] / 2

x = [-1 ± √(1 - 4 × -2)] / 2

x = [-1 ± √(1 + 8)] / 2

x = [-1 ± √(9)] / 2

x = (-1 ± 3 ) / 2

x1 = (-1 + 3 ) / 2

x1 = (2) / 2

x1 = 1

x2 = (-1 - 3 ) / 2

x2 = (-4 ) / 2

x2 = -2

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