Pass an Exam Maths

GCSE Maths · Algebra

Quadratic equations GCSE Questions and Worked Answers

A quadratic equation contains a squared unknown and can be written ax² + bx + c = 0, with a ≠ 0. For factorisable quadratics, make one side zero, factorise, and set each factor equal to zero.

Foundation & Higher6 worked examples10 original questions
Ask Ari about quadratic equationsStart the questions
Free AI tutor · GCSE quadratic equations

Practise GCSE quadratic equations for free with an AI tutor

Ask Ari for an explanation or work through an original exam-style question together.

AriYour maths coach
Chat cost ≈ $0.000000

Hi, I’m Ari. We can start quadratic equations from the beginning, work through an example together, or practise a question. Tell me which step is confusing.

Enter to send · Shift + Enter for a new line · Use $...$ or $$...$$ for math

Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.

Start with the meaning

What you need to know about quadratic equations

A rectangle is x metres wide and (x + 3) metres long. Its area is 10 m², so x(x + 3) = 10. Expanding gives x² + 3x = 10; x² means x multiplied by itself. An equation whose highest power of the unknown is 2 is quadratic. Unlike an ordinary one-step equation, it may have two solutions, so we need a method that can find both.

See the idea first

Why a zero product is useful

If two numbers multiply to zero, at least one must be zero. If neither were zero, their product could not be zero. This is what connects factorising to solving.

Make zerox² + 3x − 10 = 0subtract 10 from both sides
Factorise(x + 5)(x − 2) = 05 and −2 multiply to −10 and add to 3
Solve each conditionx = −5 or x = 2the rectangle needs x = 2 m; a width cannot be negative
From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Solve a factorisable quadratic

What the problem asks: Solve x² − 5x + 6 = 0.

How to solve it: Write (x − 2)(x − 3) = 0. Either x − 2 = 0 or x − 3 = 0, giving x = 2 or 3.

Keep a zero solution

What the problem asks: Solve x² − 4x = 0.

How to solve it: Factorise as x(x − 4) = 0. The solutions are 0 and 4. Dividing by x would lose the valid solution x = 0.

Take square roots

What the problem asks: Solve (x − 1)² = 16.

How to solve it: The bracket equals 4 or −4. Therefore x = 5 or −3. Undo the square before adding 1.

Solve when integer factors are unavailable

What the problem asks: Solve x² + x − 1 = 0 (Higher).

How to solve it: The quadratic formula gives (−1 ± √5)/2. The ± symbol asks for two calculations, one with + and one with −.

A reliable routine

For a quadratic that can be factorised

The zero-product rule applies only when the product equals zero. It does not allow you to set the factors of a product equal to a non-zero right-hand side.

  1. Rearrange so one side is zero, keeping equality balanced.
  2. Factorise fully, checking any common factor first.
  3. Set each factor equal to zero and solve the resulting linear equations.
  4. Substitute both roots into the original equation and apply any physical restrictions.

Check: A quadratic may have two distinct real solutions, one repeated real solution, or no real solutions. Factorising and the quadratic formula must give the same roots when both apply.

Fully worked

Quadratic equations GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Two bracket solutions

3 marks
Question

Solve x² + 2x − 15 = 0.

The pair 5 and −3 has product −15 and sum 2.

x2+2x−15=(x+5)(x−3)x^2+2x-15=(x+5)(x-3) (x+5)(x−3)=0(x+5)(x-3)=0

Either x + 5 = 0 or x − 3 = 0.

x=−5 or x=3x=-5\text{ or }x=3

Check: both give zero in the original expression.

Example 2

Make one side zero

3 marks
Question

Solve x² = 7x − 10.

Subtract 7x and add 10 on both sides.

x2−7x+10=0x^2-7x+10=0 (x−5)(x−2)=0(x-5)(x-2)=0 x=5 or x=2x=5\text{ or }x=2

Do not try the zero-product rule before making the right side zero.

Example 3

A common x factor

2 marks
Question

Solve x² + 6x = 0.

Both terms contain x.

x(x+6)=0x(x+6)=0 x=0 or x=−6x=0\text{ or }x=-6

Dividing by x would assume x ≠ 0 and lose a root.

Example 4

A length context

4 marks
Question

A rectangle is x cm wide and (x + 4) cm long. Its area is 21 cm². Find its dimensions.

x(x+4)=21x(x+4)=21 x2+4x−21=0x^2+4x-21=0 (x+7)(x−3)=0(x+7)(x-3)=0

The algebraic roots are −7 and 3. A length must be positive, so x = 3. The dimensions are 3 cm by 7 cm, whose product is 21 cm².

Example 5

Quadratic formula

Higher only4 marks
Question

Solve 2x² + 3x − 4 = 0. Give answers to 3 significant figures.

For ax² + bx + c = 0, use

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Here a = 2, b = 3 and c = −4.

x=−3±32−4(2)(−4)4x=\frac{-3\pm\sqrt{3^2-4(2)(-4)}}{4} =−3±414=\frac{-3\pm\sqrt{41}}4

Calculate both signs without rounding √41 first.

x≈0.851 or x≈−2.35x\approx0.851\text{ or }x\approx-2.35
Example 6

Complete the square

Higher only4 marks
Question

Solve x² + 6x + 2 = 0 exactly by completing the square.

Half of 6 is 3. Since (x + 3)² = x² + 6x + 9, replace x² + 6x by (x + 3)² − 9.

(x+3)2−9+2=0(x+3)^2-9+2=0 (x+3)2=7(x+3)^2=7 x+3=±7x+3=\pm\sqrt7 x=−3±7x=-3\pm\sqrt7

The ± includes both square roots.

10 original questions · total 29 marks

Quadratic equations GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 34 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Two roots

2 marks

Solve x² − 9 = 0.

Show worked answer
x2=9x^2=9 x=3 or x=−3x=3\text{ or }x=-3

Both signs square to 9. Alternatively, factorise as (x − 3)(x + 3) = 0 and set each factor to zero.

2

Positive constants

3 marks

Solve x² + 7x + 10 = 0.

Show worked answer
(x+5)(x+2)=0(x+5)(x+2)=0 x=−5 or x=−2x=-5\text{ or }x=-2

The factor pair adds to 7 and multiplies to 10.

3

Mixed signs

3 marks

Solve x² − x − 20 = 0.

Show worked answer
(x−5)(x+4)=0(x-5)(x+4)=0 x=5 or x=−4x=5\text{ or }x=-4

Each root makes one factor zero.

4

Rearrange

3 marks

Solve x² + 4 = 5x.

Show worked answer
x2−5x+4=0x^2-5x+4=0 (x−1)(x−4)=0(x-1)(x-4)=0 x=1 or x=4x=1\text{ or }x=4

Subtract 5x from both sides first.

5

Do not divide by x

2 marks

Solve x² − 9x = 0.

Show worked answer
x(x−9)=0x(x-9)=0 x=0 or x=9x=0\text{ or }x=9

Zero is a valid solution.

6

Repeated root

2 marks

Solve x² − 8x + 16 = 0.

Show worked answer
(x−4)2=0(x-4)^2=0 x=4x=4

Both factors give the same value, so there is one distinct root.

7

A square already isolated

3 marks

Solve (x + 2)² = 25.

Show worked answer
x+2=5 or x+2=−5x+2=5\text{ or }x+2=-5 x=3 or x=−7x=3\text{ or }x=-7

Subtract 2 in both cases.

8

Formula

Higher only4 marks

Solve x² − 2x − 2 = 0 exactly.

Show worked answer

Use a = 1, b = −2, c = −2.

x=2±(−2)2−4(1)(−2)2x=\frac{2\pm\sqrt{(-2)^2-4(1)(-2)}}2 =2±122=\frac{2\pm\sqrt{12}}2 =1±3=1\pm\sqrt3

Keep the negative coefficient in brackets when squaring it.

9

Coefficient of x² is not 1

Higher only3 marks

Solve 2x² − 5x − 3 = 0.

Show worked answer
(2x+1)(x−3)=0(2x+1)(x-3)=0 2x+1=0 or x−3=02x+1=0\text{ or }x-3=0 x=−12 or x=3x=-\frac12\text{ or }x=3

Expand the factors to check the −5x term.

10

Area and a rejected root

4 marks

A rectangle is x cm wide and (x + 2) cm long. Its area is 48 cm². Find the dimensions.

Show worked answer
x(x+2)=48x(x+2)=48 x2+2x−48=0x^2+2x-48=0 (x+8)(x−6)=0(x+8)(x-6)=0

The roots are −8 and 6. Reject the negative width, giving dimensions 6 cm by 8 cm.

Examiner-style feedback

Common quadratic equations mistakes

Not making zero first

A product equal to 12 does not mean either factor equals zero. Rearrange before using the zero-product rule.

Giving only one root

Solve both factor equations or both signs of ±, then apply any stated restrictions.

Using an unsigned coefficient

In x² − 2x − 2, b is −2 and c is −2. Include those signs in the formula.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Quadratic means highest unknown power 2.
  2. Factorise a zero expression, then solve each factor.
  3. Never divide away a possible zero root.
  4. Use context to reject impossible lengths, not algebraic roots arbitrarily.
Quick answers

Quadratic equations FAQ

When is the quadratic formula needed?

It works for every quadratic with non-zero a, including ones without convenient integer factors. It is GCSE Higher content.

What if the formula has a negative number under the square root?

There are no real solutions. At GCSE, real-number square roots cannot be taken of a negative number.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

A18: solving factorisable quadratics whose x² coefficient is 1 across tiers; other quadratic coefficients, the quadratic formula and completing the square are labelled Higher. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references