Completing the Square — GCSE Mathematics Revision
Revise Completing the Square for GCSE Mathematics. Step-by-step explanation, worked examples, common mistakes and exam-style practice aligned to AQA, Edexcel and OCR.
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Go to Simultaneous EquationsWhat is Completing the Square?
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This reveals the turning point of the quadratic graph at (-p, q). For x² + bx + c: halve the coefficient of x to get p = b/2, then write (x + b/2)² - (b/2)² + c. When a ≠ 1, factor out a first. Completing the square is also used to solve quadratic equations and derive the quadratic formula.
Step-by-step explanationWorked example
Write x² + 6x + 2 in completed square form. Half of 6 is 3. (x + 3)² = x² + 6x + 9. So x² + 6x + 2 = (x + 3)² - 9 + 2 = (x + 3)² - 7. Turning point: (-3, -7).
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Common mistakes
- 1Forgetting to subtract (b/2)² after adding it inside the square — you must compensate.
- 2Not factoring out the leading coefficient first when a ≠ 1.
- 3Getting the sign of p wrong — (x + 3)² has turning point at x = -3, not x = 3.
- 4Confusing the turning point form with the factored form — they give different information.
Completing the Square exam questions
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Step-by-step method
Step-by-step explanation
4 steps · Worked method for Completing the Square
Core concept
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This reveals the turning point of the quadratic graph at (-p, q). For x² + bx + c: halve the coefficient of x to get p = b/2, the…
Frequently asked questions
Why is completing the square useful?
It reveals the turning point of a quadratic graph without plotting, helps solve quadratics that do not factorise neatly, and is used to derive the quadratic formula.
How do I complete the square when the coefficient of x² is not 1?
Factor out the coefficient of x² first, then complete the square inside the bracket. For example, 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2(x + 2)² - 8 + 3 = 2(x + 2)² - 5.