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Edexcel IGCSE Maths (4MA1) · Unit 3: Equations & Sequences

Edexcel IGCSE 4MA1 Quadratic Equations: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

A quadratic equation has an $x^{2}$ term: rearrange it to $a x^{2} + b x + c = 0$ first. Then factorise if you can, or use the quadratic formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ , which is on the 4MA1 Higher formulae sheet. Most quadratics have two solutions; in a context, check which ones make sense.

Unit 3: Equations & Sequences
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Notes
Formulas
Worked examples
Mistakes
FAQ

Quadratic Equations notes

Solving by factorising

For $x^{2} + b x + c$ , find two numbers that multiply to $c$ and add to $b$ : $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ , so $x^{2} - 5 x + 6 = 0$ gives $x = 2$ or $x = 3$ . When $a \neq 1$ , split the middle term: $3 x^{2} + 10 x - 8 = 3 x^{2} + 12 x - 2 x - 8 = (3 x - 2) (x + 4)$ , so $x = \frac{2}{3}$ or $x = - 4$ .

The difference of two squares

$a^{2} - b^{2} = (a - b) (a + b)$ . So $x^{2} - 49 = 0$ gives $x = \pm 7$ , and $4 x^{2} - 25 = (2 x - 5) (2 x + 5)$ .

The quadratic formula

Use it when the quadratic does not factorise or the question asks for decimals. For $2 x^{2} - 3 x - 4 = 0$ : $a = 2$ , $b = - 3$ and $c = - 4$ , so $x = \frac{3 \pm \sqrt{9 + 32}}{4} = \frac{3 \pm \sqrt{41}}{4}$ , which gives $x = 2.35$ or $x = - 0.851$ (3 s.f.).

Quadratics in context

Form the equation from the question, solve it, then reject any value that does not fit, such as a negative length. A rectangle with width $x$ , length $x + 3$ and area 40 gives $x^{2} + 3 x - 40 = 0$ , so $(x + 8) (x - 5) = 0$ and the width is 5, not $- 8$ .

Key formulas

Quadratic formula, $a x^{2} + b x + c = 0$ On the formula sheet: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Expanding Learn it: $(a + b)^{2} = a^{2} + 2 a b + b^{2}, (a - b)^{2} = a^{2} - 2 a b + b^{2}, a^{2} - b^{2} = (a + b) (a - b)$

On the formula sheet = printed on the formulae sheet in the 4MA1 Higher papers, so know where to find it. Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Quadratic Equations worked examples

Worked example 1: Solving by factorising

Solve $x^{2} + 2 x - 15 = 0$ .

Solution

Two numbers multiplying to $- 15$ and adding to $2$ : $5$ and $- 3$ .
$(x + 5) (x - 3) = 0$ .
$x = - 5$ or $x = 3$ .

4 steps · 1 to fill in

Tip: Set each bracket equal to zero.

Worked example 2: Quadratic formula

Solve $x^{2} + 4 x - 7 = 0$ , giving answers to 3 significant figures.

Solution

$a = 1, b = 4, c = - 7$ .
$x = \frac{- 4 \pm \sqrt{16 + 28}}{2} = \frac{- 4 \pm \sqrt{44}}{2}$ .
$x = 1.32$ or $x = - 5.32$ .

4 steps · 1 to fill in

Tip: Brackets round negative values of b and c.

Worked example 3: Quadratic in context

The product of two consecutive positive integers is 132. Find them.

Solution

$n (n + 1) = 132$ , so $n^{2} + n - 132 = 0$ .
$(n + 12) (n - 11) = 0$ .
$n = 11$ (positive), so 11 and 12.

4 steps

Tip: Reject solutions that do not fit the context.

Play with it Roots and y-intercept: drag the sliders

Change a, p and q so that y = a(x − p)(x − q) crosses the x-axis at −1 and 3 and the y-axis at −6. Then expand your answer into the form y = ax² + bx + c.

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Quadratic Equations exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

Factorise $x^{2} + 7 x + 12$ .

Show the answer and mark scheme

B2 (x + 3)(x + 4) (B1 for (x + a)(x + b) with ab = 12 or a + b = 7)

Answer: $(x + 3) (x + 4)$

Question 2 · easy · 3 marks

Solve $x^{2} - 5 x - 14 = 0$ .

Show the answer and mark scheme

M1 (x ± 7)(x ± 2)
A1 (x − 7)(x + 2)
A1 x = 7, x = −2

Answer: $x = 7, x = - 2$

Question 3 · easy · 1 mark

Factorise $x^{2} - 81$ .

Show the answer and mark scheme

B1 (x + 9)(x − 9)

Answer: $(x + 9) (x - 9)$

Question 4 · easy · 2 marks

Factorise fully $3 x^{2} - 12$ .

Show the answer and mark scheme

B2 3(x + 2)(x − 2) (B1 for 3(x² − 4) or (3x + 6)(x − 2) oe)

Answer: $3 (x + 2) (x - 2)$

Question 5 · medium · 2 marks

Factorise $2 x^{2} + 7 x + 3$ .

Full worked solution Free, no sign-in

B2 (2x + 1)(x + 3) (B1 for factors giving two of the three terms correct)

Answer: $(2 x + 1) (x + 3)$

Every question here has a full worked solution. See them all →

Question 6 · medium · 3 marks

Solve $3 x^{2} - 10 x - 8 = 0$ .

Show the answer and mark scheme

M1 (3x ± 2)(x ± 4) or correct formula
A1 (3x + 2)(x − 4)
A1 x = 4, x = −2/3

Answer: $x = 4, x = - \frac{2}{3}$

Question 7 · hard · 3 marks

Solve $6 x^{2} + x - 12 = 0$ by factorising.

Show the answer and mark scheme

M1 (3x ± 4)(2x ± 3)
A1 (3x − 4)(2x + 3)
A1 x = 4/3, x = −3/2

Answer: $x = \frac{4}{3}, x = - \frac{3}{2}$

Question 8 · hard · 3 marks

(a) Factorise $4 x^{2} - 25 y^{2}$ . (1) (b) Hence, without expanding, find the value of $4 x^{2} - 25 y^{2}$ when $x = 15.5$ and $y = 6.2$ . (2)

Show the answer and mark scheme

B1 (a) (2x + 5y)(2x − 5y)
M1 (b) 2x = 31 and 5y = 31, so 2x − 5y = 0
A1 0

Answer: (a) $(2 x + 5 y) (2 x - 5 y)$ (b) 0

Common quadratic equations mistakes

Not rearranging to = 0 first. $x^{2} + 3 x = 10$ must become $x^{2} + 3 x - 10 = 0$ before you factorise. Writing $x (x + 3) = 10$ and setting each factor equal to 10 is wrong.
Sign slips in the formula. With $b = - 3$ , $- b = 3$ and $b^{2} = 9$ . Put negative values in brackets: $(- 3)^{2}$ , not $- 3^{2}$ .
Dividing by x. $x^{2} = 5 x$ has two solutions, $x = 0$ and $x = 5$ . Dividing both sides by $x$ loses $x = 0$ ; factorise instead: $x (x - 5) = 0$ .
Rounding too early. Keep the square root exact, or in your calculator, until the last step, then round to the accuracy the question asks for.

Where students lose marks across 4MA1, from the examiner reports →

Quadratic Equations FAQ

Is the quadratic formula on the 4MA1 formula sheet?

Yes. The quadratic formula is printed on the 4MA1 Higher formulae sheet, so you do not have to memorise it, but you do need to substitute into it carefully, with brackets round negative values.

When should I use the quadratic formula instead of factorising?

Use the formula when the question asks for an answer to a number of decimal places or significant figures, or when you cannot find the factors quickly. An instruction to round is a strong sign the quadratic does not factorise.

Can I use a calculator for quadratic equations in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Quadratic Equations revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 quadratic equations questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Quadratic Equations: exam questions and answers, notes and worked examples

Try it first

Quadratic Equations notes

Solving by factorising

The difference of two squares

The quadratic formula

Quadratics in context

Key formulas

Quadratic Equations worked examples

Worked example 1: Solving by factorising

Solution

Worked example 2: Quadratic formula

Solution

Worked example 3: Quadratic in context

Solution

Quadratic Equations exam-style questions

Question 1 · easy · 2 marks

Question 2 · easy · 3 marks

Question 3 · easy · 1 mark

Question 4 · easy · 2 marks

Question 5 · medium · 2 marks

Full worked solution Free, no sign-in

Question 6 · medium · 3 marks

Question 7 · hard · 3 marks

Question 8 · hard · 3 marks

Common quadratic equations mistakes

Quadratic Equations FAQ

Next steps

Practise quadratic equations with your progress saved

More Unit 3 topics: Equations & Sequences