Solving quadratic equations
Solving quadratic equations by factorising, completing the square, and using the quadratic formula.
Full topic guide: the detailed syllabus page with worked examples and common mistakes lives at studyvector.co.uk/gcse/maths/algebra/quadratic-equations.
Topic preview: Solving quadratic equations
Sample stems from the StudyVector question bank (AQA · Edexcel · OCR) — not generic filler text.
More questions are being linked to this topic. You can still start low-focus cards after you create a free account.
Curated launch topic
This is one of the first GCSE Mathematics guides we are pushing deepest
High-intent GCSE Maths pages built around percentage change, algebra, trigonometry, geometry, probability, and statistics routes where method marks are most often lost. This page focuses on Move between factorising, solving, and checking roots without treating every quadratic like the same routine., then hands you into practice instead of leaving you on a dead-end revision article.
Coverage and provenance
What this page is based on
StudyVector does not present unsupported question coverage as complete. Read how questions are selected and reviewed.
Topic explanation
A quadratic equation has the form ax² + bx + c = 0. You can solve quadratics by factorising, using the quadratic formula x = (-b ± √(b²-4ac)) / 2a, or completing the square. Factorising is fastest when it works, but the formula always works. The discriminant b²-4ac tells you how many solutions exist: positive means two, zero means one (repeated), negative means none (no real roots).
Solving quadratic equations 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 Solving quadratic equations 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 Solving quadratic equations 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 Solving quadratic equations 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 Solving quadratic equations 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 Solving quadratic equations, 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
Solve x² - 5x + 6 = 0. Factorise: (x - 2)(x - 3) = 0. So x = 2 or x = 3.
Example 2
Identify the task before answering
Question type: a Solving quadratic equations prompt asks for a clear response in GCSE Mathematics. Step 1: underline the command word. Step 2: name the exact part of Solving quadratic equations 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: Solving quadratic equations improves faster when feedback creates a specific retry, not another passive reading session.
Stay inside this launch cluster
These are the other high-intent GCSE Mathematics topic guides we are shaping first. Use them when you want a stronger next page than a generic topic list.
Number
Percentage Change & Reverse Percentages
Separate multiplier method, percentage increase/decrease, and reverse-percentage logic so common calculator slips stop costing easy marks.
Algebra
Simultaneous Equations
Choose elimination or substitution cleanly and keep each algebra step visible so solutions stop collapsing mid-method.
Geometry & Measures
SOHCAHTOA Problems
Pick the right trig ratio from the diagram, set it up carefully, and keep calculator and rounding control under pressure.
Probability
Tree Diagrams
Turn conditional and multi-step probability into one reliable branch method instead of guessing which events to multiply or add.
Next revision routes from this subject
Good topic pages should lead naturally into the next useful page. Use these links to stay inside the same strand or jump into the next topic area without starting your search again.
Stay in the same topic area
Common mistakes
- Forgetting to rearrange the equation to = 0 before factorising.
- Sign errors in the quadratic formula, especially with the -b term when b is already negative.
- Giving only one solution when there are two (forgetting the ± in the formula).
- Not checking whether the question asks for exact answers (surds) or decimal approximations.
Exam board notes
All boards require factorising and the quadratic formula at Higher. Completing the square is also Higher content. AQA sometimes asks students to derive the formula.
FAQs
When should I use the quadratic formula instead of factorising?
Use the formula when the quadratic does not factorise neatly, or when the question specifically asks for answers to a given number of decimal places or significant figures.
What does the discriminant tell you?
The discriminant is b² - 4ac. If it is positive, there are two distinct real roots. If zero, there is one repeated root. If negative, there are no real roots.
More on StudyVector
Full practice set
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