Completing the square

Completing the square writes a quadratic as (x + p)² + q. It shows the turning point straight away, solves any quadratic, and is how the quadratic formula is proved.

Key facts

  • x² + bx + c = (x + b/2)² − (b/2)² + c
  • The turning point of y = (x + p)² + q is (−p, q).
  • If there’s a number in front of x², take it out as a factor first.

Worked examples

Example 1

Write x² + 6x + 2 in the form (x + p)² + q.

  1. Halve 6 to get 3. (x + 3)² = x² + 6x + 9
  2. We have 2, not 9, so subtract 7. x² + 6x + 2 = (x + 3)² − 7

Answer: (x + 3)² − 7

Example 2

Solve x² − 4x − 1 = 0, leaving surds.

  1. (x − 2)² − 4 − 1 = 0 (x − 2)² = 5
  2. Square root both sides. x − 2 = ±√5
  3. x = 2 ± √5

Answer: x = 2 ± √5