Find complex solutions of a quadratic equation

Learn how to find complex solutions of a quadratic equation with these two examples.

Example #1:

Solve 8x² + 200 = 0

Subtract 200 from each side of the equation

8x² + 200 - 200 = 0 - 200

8x² = -200

Divide each side of the equation by 8

(8x²)/8 = -200/8

x² = -25

x = ±√(-25)

x = ±√[(25)×-1]

x = ±[√(25) × √(-1)]

x = ±(5 × i)

x = ±5i

x₁ = 5i

x₂ = -5i

Check for 5i

8(5i)² + 200 = 0

8(25i²) + 200 = 0

8(-25) + 200 = 0

-200 + 200 = 0

0 = 0

Check for -5i

8(-5i)² + 200 = 0

8(25i²) + 200 = 0

8(-25) + 200 = 0

-200 + 200 = 0

0 = 0

Example #2:

Solve x² + 4x + 5 = 0

Let us solve by completing the square

Solve x² + 4x + 5 = 0

Isolate the quadratic and the linear term.

x² + 4x + 5 - 5 = 0 - 5

x² + 4x  = -5

Complete the square

x² + 4x + 2² = -5 + 2²

(x + 2)² = -5 + 4

(x + 2)² = -1

x + 2 = ±i

Subtract 2 from each side of the equation

x + 2 - 2 = ±i - 2

x = ±i - 2

x₁ = i - 2

x₂ = -i - 2

Check for i - 2

(i - 2)² + 4(i - 2) + 5 = 0

Factor out (i - 2)

(i - 2)[(i - 2) + 4] + 5 = 0

(i - 2)(i + 2) + 5 = 0

i² - 2² + 5 = 0

-1 - 4 + 5 = 0

-5 + 5 = 0

0 = 0

Check for -i - 2

(-i - 2)² + 4(-i - 2) + 5 = 0

Factor out (-i - 2)

(-i - 2)[(-i - 2) + 4] + 5 = 0

(-i - 2)(-i + 2) + 5 = 0

(-i)² - 2² + 5 = 0

(i)² - 2² + 5 = 0

-1 - 4 + 5 = 0

-5 + 5 = 0

0 = 0