Quadratic equations
A quadratic equation has a non-zero squared term as its highest power. Put it in the form ax² + bx + c = 0 before choosing a method. Factorisation is convenient when integer factors are clear; the formula also handles roots that do not factor neatly.
What this guide covers
Solving by factorisation, the quadratic formula, completing the square and graphs. The examples sample algebraic methods; a graph gives approximate roots unless exact values can be established.
Syllabus reference: 2026 Mathematics 4052, N7, page 7. These are Mentora search groupings, not new official syllabus categories.
Worked example
Solve x² − 8x + 15 = 0.
The factors are (x − 3)(x − 5). A product equals zero only if at least one factor is zero, so x = 3 or x = 5. Substitution gives 9 − 24 + 15 = 0 and 25 − 40 + 15 = 0.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Solve x² = 49.
Show worked answer
Both 7² and (−7)² equal 49, so x = 7 or x = −7.
Check 2
Solve 2x² + x − 3 = 0.
Show worked answer
Factor as (2x + 3)(x − 1) = 0. Thus x = −3/2 or x = 1.
Check 3
Solve x² − 4x + 1 = 0 exactly.
Show worked answer
Complete the square: (x − 2)² − 3 = 0. Thus x − 2 = ±√3 and x = 2 ± √3.
Common mistakes to watch for
- Move all terms to one side before applying the zero-product rule.
- Do not lose the negative root when taking square roots.
- In the quadratic formula, the denominator 2a divides the entire numerator. Keep signed values of a, b and c in brackets.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
You may need first
These verified prerequisite links suggest useful foundations. They do not make a claim about your current readiness.
- Factorisation and subject of formulaFactorisation lets you use the zero-product rule once the equation equals zero.Practise factorisation →
- Quadratic functions and their graphsA quadratic graph shows how equation solutions correspond to x-axis intersections. This is especially useful for the graphical method.Practise functions and graphs →
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More G3 Math topicsPublished by Mentora · Sources checked 11 Sept 2026
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Original public examples and answers were checked with automated mathematical/editorial QA. No individual author, human reviewer or independent expert review is recorded.