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.
Full topic guide: the detailed syllabus page with worked examples and common mistakes lives at studyvector.co.uk/gcse/maths/algebra/completing-the-square.
Topic preview: Completing the Square
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Topic explanation
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.
Completing the Square is easiest to revise when it is treated as a precise exam behaviour, not a loose note-taking category. In GCSE Mathematics, the goal is to recognise how the topic appears in a question, identify the command word, and decide what evidence, method, or vocabulary earns marks. StudyVector keeps this page tied to AQA · Edexcel · OCR language where coverage is available, then routes practice towards the same topic so revision moves from explanation into retrieval.
A strong revision session starts with a short recall check. Write down the rule, definition, process, or method linked to Completing the Square before looking at any notes. Then answer one exam-style prompt and compare your answer with the mark-scheme logic: did you make a clear point, support it with the right step, and avoid drifting into a nearby topic? This matters because many lost marks come from almost-correct answers that do not match the expected structure.
Use this guide as the first layer: understand the topic, look at the worked examples, complete the mini quiz, then move into full practice. The full StudyVector practice loop is designed to capture whether mistakes are caused by knowledge, method, language, or timing. That distinction is important. If the error is factual, you need reteaching. If the error is method-based, you need a worked retry. If the error is wording, you need command-word calibration. That is how Completing the Square becomes a controlled revision target rather than another page in a folder.
Lost marks → repair task
Why marks are usually lost here
These are the error patterns StudyVector looks for after an attempt. The goal is not a generic explanation; it is one repair move and one follow-up question.
Unit, formula, or method slip
Examiner move: Select the correct method and keep units, substitutions, signs, and rounding visible.
Repair drill: Redo the calculation or method line slowly, naming the formula before substituting values.
Missing chain of reasoning
Examiner move: Show the link between point, method, evidence, and conclusion instead of jumping to the final line.
Repair drill: Write the missing because/therefore step, then retry one isomorphic question.
Timing breakdown
Examiner move: Match answer length to marks and avoid over-writing low-mark questions.
Repair drill: Set a one-mark-per-minute cap and write a compact version before expanding.
Mini quiz
Use these checks before full practice. They test topic recognition, exam technique, and whether you can connect the explanation to a marked response.
1. What should you check first when a Completing the Square question appears in GCSE Mathematics?
- A.The command word and the exact topic focus
- B.The longest paragraph in your notes
- C.A memorised answer from a different topic
2. Which revision action gives the strongest evidence that Completing the Square is improving?
- A.Rereading the explanation twice
- B.Answering a timed exam-style question and reviewing lost marks
- C.Highlighting every key phrase in the topic notes
Sample questions
Topic-specific public question previews are still being reviewed. We keep them off public pages until the topic match is safe.
Exam tips
- Read the command word carefully — "explain" needs reasons; "state" expects a short fact.
- For Completing the Square, show structured working even when you are practising multiple choice — it builds accuracy under time pressure.
- Mark yourself against the mark scheme style: one clear point per mark, in logical order.
- Come back to this topic after a day or two; short spaced reviews beat one long cram.
Worked examples
Example 1
Modelled exam response
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).
Example 2
Identify the task before answering
Question type: a Completing the Square prompt asks for a clear response in GCSE Mathematics. Step 1: underline the command word. Step 2: name the exact part of Completing the Square being tested. Step 3: decide whether the mark scheme wants a definition, method, explanation, comparison, or calculation. Why it works: most weak answers fail before the content starts because they answer the topic generally rather than the exact exam task.
Example 3
Turn feedback into a repair task
Suppose your answer shows partial understanding but loses marks for precision. First, rewrite the missing mark as a short target: "I need to state the mechanism, unit, reason, or evidence explicitly." Then answer one similar question without notes. Finally, compare the second attempt with the first and check whether the same mark was recovered. Why it works: Completing the Square improves faster when feedback creates a specific retry, not another passive reading session.
Next revision routes from this subject
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Common mistakes
- Forgetting to subtract (b/2)² after adding it inside the square — you must compensate.
- Not factoring out the leading coefficient first when a ≠ 1.
- Getting the sign of p wrong — (x + 3)² has turning point at x = -3, not x = 3.
- Confusing the turning point form with the factored form — they give different information.
FAQs
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.
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