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AQA GCSEAQA GCSE Maths Walkthrough: Solving a Quadratic by Completing the Square
Completing the square appears on both Higher tier papers and is a reliable 3-4 mark question. This walkthrough shows the full method step by step and flags the sign error that costs most students their marks.
The question
Solve x² + 6x + 7 = 0 by completing the square. Give your answers in the form a ± b√c where a, b, and c are integers. [4 marks]
[4 marks]
Full worked solution
Write the completed square form
Take the coefficient of x (which is 6), halve it (giving 3), and write: x² + 6x + 7 = (x + 3)² − 9 + 7 The −9 comes from subtracting the square of 3: we added 3² = 9 when we formed (x + 3)², so we must subtract it back. Simplify: (x + 3)² − 2 = 0
Rearrange to isolate the bracket
(x + 3)² = 2 Square root both sides: x + 3 = ±√2 IMPORTANT: You must write ±√2 — both the positive and negative root. Missing the ± loses a mark.
Solve for x
x = −3 ± √2 This gives two solutions: x = −3 + √2 ≈ −1.59 x = −3 − √2 ≈ −4.41 The answer in the required form is x = −3 ± √2, where a = −3, b = 1, c = 2.
The mark scheme breakdown
- M1: correct completed square form (x + 3)² seen
- M1: correct rearrangement to (x + 3)² = 2
- A1: x = −3 ± √2 (both solutions, with ± present)
- A1: final answer in correct a ± b√c form
The three mistakes that cost marks
Mistake 1 — wrong sign on the constant: Students write (x + 3)² + 9 + 7 instead of (x + 3)² − 9 + 7. Remember: completing the square adds the term (b/2)² inside the bracket, so you must subtract it from the outside. Mistake 2 — forgetting ±: Writing x + 3 = √2 instead of ±√2 gives only one solution and loses the accuracy mark. Mistake 3 — arithmetic error combining constants: −9 + 7 = −2, not +2. A sign error here produces x + 3 = ±√2 (correct form but wrong value) and loses the final accuracy mark.
When completing the square appears on the paper
Completing the square typically appears in the later questions of Paper 2 or Paper 3 (both calculator papers) on Higher tier. It is often a 4-mark question asking for solutions "in the form a ± b√c" or "to 2 decimal places". If the question says "by completing the square", you must use that method — using the quadratic formula instead will not score method marks even if the answer is correct. If the question says "solve the quadratic", you may choose any valid method.
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