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 in the standard of a quadratic equation


Quadratic Formula is x = (-b ± √(b² - 4ac))/2a


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


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