GCSE specification fit
A major algebra skill for Foundation and Higher tier.
Quadratics are expressions with a squared term, such as x² + 5x + 6. GCSE questions may ask you to expand brackets, factorise into brackets, solve an equation, choose a higher-tier method, or connect the answers to a quadratic graph.
What you will learn
Why this matters
Quadratic algebra turns up in graphs, area problems, sequences, formulae and higher-tier problem solving. The same bracket skills also help with later algebraic fractions and proof.
The key habit is to keep expressions and equations separate: factorising an expression rewrites it, while solving an equation finds values of x.
Prior knowledge
You should already be comfortable with:
Clear explanation
What makes an expression quadratic?
A quadratic expression has a squared variable as its highest power. In GCSE Maths this is usually x², but it could be another letter such as n² or y².
Expanding double brackets
Multiply every term in the first bracket by every term in the second bracket, then collect like terms.
Factorising reverses expanding
To factorise x² + 7x + 10, look for two numbers that multiply to 10 and add to 7. They are 2 and 5.
Solving means make the expression equal to zero
If two brackets multiply to make zero, at least one bracket must be zero. This is called the zero-product rule.
Worked examples
Example 1: Expand double brackets
Expand (x + 3)(x + 4).
x² + 4x + 3x + 12 x² + 7x + 12Example 2: Factorise a simple quadratic
Factorise x² + 9x + 20.
Find two numbers that multiply to 20 and add to 9. 4 and 5 work.Example 3: Solve by factorising
Solve x² − 3x − 10 = 0.
x² − 3x − 10 = (x − 5)(x + 2) x − 5 = 0 or x + 2 = 0 x = 5 or x = −2Example 4: Rearrange first
Solve x² + 6x = 16.
x² + 6x − 16 = 0 (x + 8)(x − 2) = 0 x = −8 or x = 2Quick checks
Choose an answer, then check your thinking.
1. Which expression is quadratic?
2. What is (x + 1)(x + 6) when expanded?
3. If (x − 4)(x + 3) = 0, what are the solutions?
Practice questions
Question 1
Expand (x + 2)(x + 7).
Reveal answer and marking guidance
Answer: x² + 9x + 14.
Marking: Show four terms first or clearly collect the middle terms 7x + 2x = 9x.
Question 2
Expand and simplify (x − 3)(x + 8).
Reveal answer and marking guidance
Answer: x² + 5x − 24.
Marking: Multiply to get x² + 8x − 3x − 24, then collect to x² + 5x − 24.
Question 3
Factorise x² + 8x + 15.
Reveal answer and marking guidance
Answer: (x + 3)(x + 5).
Marking: Use two numbers that multiply to 15 and add to 8.
Question 4
Factorise x² − x − 12.
Reveal answer and marking guidance
Answer: (x − 4)(x + 3).
Marking: Use two numbers that multiply to −12 and add to −1: −4 and 3.
Question 5
Solve x² + 6x + 8 = 0.
Reveal answer and marking guidance
Answer: x = −2 or x = −4.
Marking: Factorise to (x + 2)(x + 4) = 0, then set each bracket equal to zero.
Question 6
Solve x² − 2x − 15 = 0.
Reveal answer and marking guidance
Answer: x = 5 or x = −3.
Marking: Factorise to (x − 5)(x + 3) = 0, then solve x − 5 = 0 and x + 3 = 0.
Question 7
Solve x² + 4x = 12.
Reveal answer and marking guidance
Answer: x = 2 or x = −6.
Marking: Rearrange first: x² + 4x − 12 = 0. Factorise to (x + 6)(x − 2) = 0, then set each bracket equal to zero.
Question 8
Solve 2x² + 5x − 3 = 0 by factorising.
Reveal answer and marking guidance
Answer: x = 1/2 or x = −3.
Marking: Factorise to (2x − 1)(x + 3) = 0. Then 2x − 1 = 0 gives x = 1/2, and x + 3 = 0 gives x = −3.
Question 9
Solve x² − 4x − 1 = 0 using the quadratic formula. Give your answers in exact form.
Reveal answer and marking guidance
Answer: x = 2 + √5 or x = 2 − √5.
Marking: Use a = 1, b = −4 and c = −1. The formula gives x = (4 ± √20) ÷ 2, which simplifies to 2 ± √5.
Question 10
A rectangle has length x + 5 cm and width x + 2 cm. Its area is 60 cm². Form and solve a quadratic equation to find x.
Reveal answer and marking guidance
Answer: x = 5.
Marking: (x + 5)(x + 2) = 60, so x² + 7x + 10 = 60 and x² + 7x − 50 = 0. Factorise to (x + 10)(x − 5) = 0, giving x = −10 or x = 5. Only x = 5 fits a rectangle with positive side lengths.
Answers and marking guidance
The exact practice answers are hidden under each question so you can try first. For quadratic work, marks usually come from expanding every bracket term, collecting like terms accurately, finding factor pairs with the correct sum, setting the equation equal to zero before solving, and giving both solutions. When factorising does not work neatly, write a, b and c carefully in the quadratic formula and simplify exact surd answers only when the square-root factor allows it.
Common mistakes
- Missing the middle terms: double brackets usually create two x terms before they are collected.
- Solving an expression: you can only solve when there is an equals sign.
- Forgetting to make zero: factorising solves a quadratic equation only after one side is 0.
- Giving one solution: most factorised quadratics give two possible x values.
- Sign errors: negative factors affect both the middle term and the final number.
Extension challenge
Solve 2x² + 7x + 3 = 0 by factorising.
Reveal answer
Answer: x = −3 or x = −1/2.
Factorise as (2x + 1)(x + 3) = 0. Then 2x + 1 = 0 gives x = −1/2, and x + 3 = 0 gives x = −3.
Exam-board guidance
Quadratic expressions and equations are assessed by every GCSE Maths board. Foundation questions tend to focus on expanding, simple factorising and solving by factors; Higher questions may also use harder factorising, completing the square, the quadratic formula and graph links.
AQA GCSE Maths
Make the equation equal to zero, show your factorising step before giving the two solutions, and check both values by substitution if time allows.
OCR GCSE Maths
Keep the equation equal to zero before using factors, otherwise the zero-product rule does not apply; include both roots unless the context rules one out.
Pearson Edexcel GCSE Maths
Questions may mix expanding, simplifying and factorising before asking you to solve, so collect everything to one side before using the zero-product rule.
Eduqas GCSE Maths
Write the two bracket factors carefully, check their product and sum, then turn each bracket into a simple equation with clear signs.
WJEC Wales
Expect quadratic work to appear as algebra practice and sometimes inside graph or modelling questions, where both roots must be checked against the context.
CCEA GCSE Maths
Be ready to show algebraic working clearly, especially in non-calculator units where factorising, sign handling and both roots earn method marks.
Next lesson
Next, move into Quadratic Graphs.