Lesson overview
A major algebra skill for Core and extension practice 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 more demanding method, or connect the answers to a quadratic graph.
Quadratic expressions and equations infographic
Before viewing
Before viewing, identify whether the example is expanding, factorising or solving.
After viewing
After viewing, explain why a solved quadratic can have two solutions.
What you will learn
Key facts before you start
Why this matters
Quadratic algebra turns up in graphs, area problems, sequences, formulae and more demanding 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 + 12Reveal answer
Answer: x² + 7x + 12.
Example 2: Factorise a simple quadratic
Factorise x² + 9x + 20.
Find two numbers that multiply to 20 and add to 9. 4 and 5 work.Reveal answer
Answer: (x + 4)(x + 5).
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 = −2Reveal answer
Answer: x = 5 or x = −2.
Example 4: Rearrange first
Solve x² + 6x = 16.
x² + 6x − 16 = 0 (x + 8)(x − 2) = 0 x = −8 or x = 2Reveal answer
Answer: x = −8 or x = 2.
Quick 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. Core questions tend to focus on expanding, simple factorising and solving by factors; Extension questions may also use harder factorising, completing the square, the quadratic formula and graph links.
AQA GCSE Maths
Focus on the shared method, notation and checking habit; wording and context can vary by route.
OCR GCSE Maths
Show clear method steps and use precise notation; question wording may vary by route.
Pearson Edexcel GCSE Maths
Keep working visible and check units or notation; contexts may vary by route.
Eduqas GCSE Maths
Use the same core skill in practical contexts, with clear reasoning and final checks.
WJEC Wales
Connect the method to the context and state what the result means.
CCEA GCSE Maths
Make each method step visible and keep notation, units and final checks clear.
Next lesson
Next, move into Quadratic Graphs.