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, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), SQA, IB, AP.
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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).
Mini lesson for Completing the Square
1. Understand the core idea
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).
Can you explain Completing the Square without copying the notes?
2. Turn it into marks
Write x² + 6x + 2 in completed square form. Half of 6 is 3.
Underline the method, evidence, or command-word move that would earn credit in GCSE Algebra.
3. Fix the likely mark leak
Watch for this mistake: Forgetting to subtract (b/2)² after adding it inside the square — you must compensate.
Write one correction rule before doing another practice question.
Practise this topic
Jump into adaptive, exam-style questions for Completing the Square. Free to start; sign in to save progress.
Completing the Square practice questions
These are original StudyVector questions for revision practice. They are not official exam-board questions.
Question 1
In one GCSE sentence, explain what Completing the Square is testing.
Answer: 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).
Mark focus: Precise definition and topic focus.
Question 2
A student sees a Completing the Square question but is not sure how to start. What should the first method line establish?
Answer: It should identify the rule, equation, diagram feature, or transformation before any calculation. That protects method marks and makes later checking easier.
Mark focus: Method selection and command-word control.
Question 3
A student makes this mistake: "Forgetting to subtract (b/2)² after adding it inside the square — you must compensate." What should their next repair task be?
Answer: Do one Completing the Square question and review the mistake type.
Mark focus: Error correction and next-step practice.
Completing the Square flashcards
Core idea
What is the main idea in 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).
Common mistake
What mistake should you avoid in Completing the Square?
Forgetting to subtract (b/2)² after adding it inside the square — you must compensate.
Practice
What is one useful practice task for Completing the Square?
Answer one Completing the Square question and review the mistake type.
Exam board
How should you use board notes for Completing the Square?
Use your own GCSE specification for exact paper wording and depth.
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
Exam-style questions for Completing the Square with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), SQA, IB, AP specifications.
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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.