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Algebra · Higher

Completing the square

Completing the square rewrites a quadratic in the form (x + p)² + q. This reveals the minimum (or maximum) point and can be used to solve quadratic equations.

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Key facts to remember

  • 1x² + bx = (x + b/2)² − (b/2)²
  • 2For y = x² + bx + c: completing the square gives y = (x + b/2)² + (c − (b/2)²).
  • 3The minimum point (vertex) is at (−b/2, c − (b/2)²).
  • 4To solve, rearrange so (x + p)² = k, then x = −p ± √k.
  • 5For ax² + bx + c, first factor out a before completing the square.
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Formulas

Completing the square
x² + bx + c = (x + b/2)² − (b/2)² + c
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Worked examples

Example 1

Write x² − 6x + 11 in completed square form and find the minimum point.

Working

  1. Half of −6 is −3
  2. (x − 3)² = x² − 6x + 9
  3. x² − 6x + 11 = (x − 3)² − 9 + 11 = (x − 3)² + 2
  4. Minimum point: (3, 2)
Answer(x − 3)² + 2; minimum at (3, 2)
Example 2

Solve x² + 4x − 3 = 0 by completing the square, giving answers in surd form.

Working

  1. (x + 2)² − 4 − 3 = 0
  2. (x + 2)² = 7
  3. x + 2 = ±√7
  4. x = −2 + √7 or x = −2 − √7
Answerx = −2 ± √7
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Common mistakes

✗Halving the coefficient of x² instead of the coefficient of x.
✗Forgetting to subtract (b/2)² after writing the squared bracket.
✗Sign errors: for (x − 3)², the vertex is at x = +3, not x = −3.
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Exam tips

✓Always halve the x-coefficient to find p, then subtract p² to compensate.
✓Check by expanding your completed square form — it should give the original quadratic.

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