Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want quadratic equations and inequalities — factorising, the formula, completing the square and sketching — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise quadratic equations and inequalities in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the quadratic revision-guide angle, while Tutopiya’s Quadratic Equations and Inequalities subtopic page owns the learning resource and the free quadratic quiz owns the practice.

Quadratic equations and inequalities sit at the heart of the Algebra unit in Cambridge IGCSE Mathematics (0580/0607). Examiners rarely ask a single isolated skill — they combine factorising, the quadratic formula, completing the square and inequality notation in questions worth three to six marks. This guide explains exactly what the subtopic covers, how to decode the command words that appear on papers, and where to practise each method until it is automatic.

Key takeaways

• A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0; a quadratic inequality uses <, >, ≤ or ≥ instead of =.
• Factorise first when the trinomial factorises cleanly; use the quadratic formula when it does not.
• Completing the square reveals the turning point and is often required when the question says “write in the form (x + p)² + q”.
• Sketch questions reward roots, y-intercept and turning point — not a perfect artistic curve.

What are quadratic equations and inequalities in Cambridge IGCSE Maths?

A quadratic equation is an equation where the highest power of the unknown is 2. In Cambridge IGCSE Mathematics it appears as ax² + bx + c = 0, and you must solve for x using factorisation, the quadratic formula or completing the square. Quadratic inequalities extend the same expressions with inequality symbols; the solution is usually a range of x-values, not a single pair of roots.

Read the full explanation and worked examples on Tutopiya’s Quadratic Equations and Inequalities subtopic page before you attempt questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Factorisation	Write ax² + bx + c as (px + q)(rx + s) = 0	”Solve x² − 5x + 6 = 0 by factorising”
Quadratic formula	x = (−b ± √(b² − 4ac)) / 2a	”Solve 2x² + 3x − 7 = 0. Give your answers correct to 2 decimal places.”
Completing the square	Rewrite as a(x + p)² + q	”Express x² − 6x + 1 in the form (x − p)² + q”
Discriminant b² − 4ac	Tells you how many real roots exist	”Show that 3x² + 2x + 5 = 0 has no real solutions”
Quadratic inequality	Find the x-values that satisfy ax² + bx + c > 0	”Solve x² − 4x − 5 > 0”

How to solve a quadratic equation — step by step

1. Rearrange to ax² + bx + c = 0 if needed. Example: x² = 3x + 4 becomes x² − 3x − 4 = 0.
2. Try factorisation when a = 1 or when you spot a common factor. Set each bracket equal to zero.
3. If factorisation fails, use the quadratic formula. Substitute a, b and c carefully; watch the sign of b.
4. For “correct to n decimal places”, use your calculator and round only at the end.
5. Sanity-check: substitute each root back into the original equation.

Test whether the method has stuck with the free Quadratic Equations quiz.

Quadratic inequalities: which region satisfies the inequality?

After finding the critical values (the roots), sketch a quick number line or parabola:

1. Solve the related equation ax² + bx + c = 0 to find critical values.
2. Sketch or test a value in each region between the roots.
3. For > 0, choose regions where the curve is above the x-axis; for < 0, below.
4. Write the answer as an inequality or interval notation as the question demands.

Quadratic equations in past-paper wording: command words that matter

Most lost marks come from misreading the command word — the instruction that tells you exactly what to do.

Command word / phrase	What the question wants	Typical quadratic stem
Solve … by factorising	Factorise only — formula loses method marks	”Solve x² + 7x + 12 = 0 by factorising.”
Solve … giving your answers correct to … decimal places	Quadratic formula or calculator; round at end	”Solve 3x² − 2x − 1 = 0. Give your answers correct to 2 decimal places.”
Express … in the form (x + p)² + q	Completing the square	”Express x² + 8x − 3 in the form (x + p)² + q.”
Show that	Prove the given result — answer is stated	”Show that x² − 4x + 5 = (x − 2)² + 1.”
Sketch the graph of	Axes, roots, y-intercept, turning point	”Sketch the graph of y = x² − 4x − 5.”
Solve the inequality	Critical values plus correct region	”Solve x² − 2x − 8 ≤ 0.”

Worked exam-style stems (how to answer the wording)

Practising the wording — not just the maths — is what method marks reward.

1. “Solve x² − 9x + 20 = 0 by factorising.” (x − 4)(x − 5) = 0 → x = 4 or x = 5. Mark-scheme reward: correct factors, then both roots.
2. “Solve 2x² − 3x − 5 = 0. Give your answers correct to 2 decimal places.” Formula: a = 2, b = −3, c = −5 → x = 2.05 or x = −1.22 (2 d.p.). Reward: correct substitution into the formula, then rounding.
3. “Show that the equation 2x² + 4x + 7 = 0 has no real solutions.” Discriminant: b² − 4ac = 16 − 56 = −40 < 0 → no real roots. Reward: working for the discriminant, then the conclusion.

When you can recognise the wording instantly, work the full set on the Algebra topical past-paper questions and the Quadratic Equations quiz.

How quadratics connect to the rest of Algebra and Functions

Factorising quadratics feeds directly into Factorisation, and completing the square links to Quadratic Functions. When you are ready to mix topics, the Cambridge IGCSE Maths resource hub lets you move from a weak subtopic straight into the next.

Common mistakes students make

• Using the quadratic formula when the question says “by factorising” — you lose method marks.
• Forgetting the ± in the formula and only writing one root.
• Solving inequalities by writing x = … instead of a range of values.
• Sketching without labelling roots, y-intercept or turning point when the mark scheme expects them.
• Sign errors when rearranging to ax² + bx + c = 0.

When you need more support

If quadratic questions keep tripping you up — especially completing the square or inequalities — work through the Factorisation quiz and the Algebra topical past-paper questions, then get focused help from a Cambridge IGCSE Maths tutor.

Frequently asked questions

Is the quadratic formula given in the exam? Yes — Cambridge IGCSE Mathematics provides the quadratic formula on the formula sheet. You still need to substitute a, b and c correctly and show your working when marks are available.

When should I factorise instead of using the formula? Factorise when the question says so, or when the trinomial factorises cleanly in a few steps. Use the formula when factorisation is awkward or when the question asks for decimal answers.

How do I solve x² − 4x − 5 > 0? Solve x² − 4x − 5 = 0 to get x = −1 and x = 5. The parabola opens upward, so the expression is positive when x < −1 or x > 5.

How do I revise quadratic equations effectively? Read the subtopic notes, work examples by hand for each method, then take the quadratic quiz. Revisit any inequality or completing-the-square questions you got wrong before moving on.

Ready to master Cambridge IGCSE Maths quadratics?

Start with the Quadratic Equations and Inequalities subtopic page, then book a free trial with a Cambridge IGCSE Maths specialist to turn quadratics into guaranteed marks.

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