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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.

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.

Useful foundations to check

These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.

Applications and next topics

Roots locate x-axis intersections of quadratic graphs. In length or time problems, substitute back and reject roots that are impossible in the stated context, not simply all negative roots.

Choose a useful next resource

To understand a recurring error rather than relearn the whole skill: Connect equation roots with graph errors.

Explore the foundations connected to quadratic equations

Sources and review

Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-11; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.