Solving quadratic equations by completing the square

We can solve quadratic equations by completing the square.

Steps

1. Make the quadratic in the form ax^2 + bx + c = 0 if it isn’t already.
2. If a \neq 1, divide the whole equation by a to get rid of the coefficient of x^2.
3. Move the constant term c to the other side of the equation.
4. Complete the square on the left side of the equation using the method from the last page.
5. Move any constants from the left side to the right side by adding or subtracting.
6. Take the square root of both sides (remembering to include both the positive and negative roots).
7. Solve for x by moving any constants to the right side.

Examples

Example: Solve x^2 + 6x + 5 = 0 by completing the square

• Move the constant to the other side:
  • x^2 + 6x = -5
• Complete the square on the left side:
  • (x^2 + 6x) = -5
  • (x + 3)^2 - 9 = -5
  • (x + 3)^2 = 4
• Take the square root of both sides:
  • x + 3 = \pm 2
• Solve for x:
  • x = -3 \pm 2
  • x = -1 or x = -5

Answer: x = -1 or x = -5

Example: Solve 2x^2 + 8x + 6 = 0 by completing the square

• Divide the whole equation by 2:
  • x^2 + 4x + 3 = 0
• Move the constant to the other side:
  • x^2 + 4x = -3
• Complete the square on the left side:
  • (x^2 + 4x) = -3
  • (x + 2)^2 - 4 = -3
  • (x + 2)^2 = 1
• Take the square root of both sides:
  • x + 2 = \pm 1
• Solve for x:
  • x = -2 \pm 1
  • x = -1 or x = -3

Answer: x = -1 or x = -3

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