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Completing the square practice questions

Completing the square rewrites a quadratic as (x plus a) squared plus b. Halve the x coefficient to get a, then subtract a squared and add the constant to find b. For x squared plus 6x plus 11, the answer is (x plus 3) squared plus 2, which also reveals the turning point.

Completing the square rewrites a quadratic as (x + a)² + b. It is a GCSE Higher topic that reveals the turning point of a parabola immediately, and it is the method from which the quadratic formula itself is derived.

GCSE HigherA-level

The method, step by step

  1. Halve the x coefficient. This value becomes the number inside the bracket.
  2. Square the bracket mentally. Expanding (x + a)² produces an extra a² you did not want.
  3. Subtract the excess. Take away a², then add the original constant.
  4. Simplify. Combine the two numbers outside the bracket.

Worked example

Write x² + 6x + 11 in the form (x + a)² + b
Halve the x coefficient: 6 ÷ 2 = 3, so the bracket is (x + 3).
(x + 3)² expands to x² + 6x + 9, which is 9 more than wanted.
Correct it: 11 − 9 = 2.
x² + 6x + 11 = (x + 3)² + 2

Common mistakes

Forgetting to subtract the square of the halved number, which leaves the expression unequal to the original.

Halving the constant instead of the x coefficient.

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