GCSE Maths Higher: Algebra practice questions
Ten free GCSE Maths Higher tier questions on quadratics, simultaneous equations, algebraic fractions, completing the square, functions and quadratic sequences, each with a worked solution. Suitable for AQA, Edexcel and OCR.
Original questions written by Studoli tutors in the style of the GCSE exam. They are not official past questions.
Timed mode
10 questions in 15 minutes, one question at a time. Your answers are marked when you submit or when the time runs out.
Topic: Factorising quadratics
1. Factorise x² + 5x + 6.
Solution
B. (x + 2)(x + 3)
- Find two numbers that multiply to 6 and add to 5: 2 and 3.
- x² + 5x + 6 = (x + 2)(x + 3).
Topic: Quadratic equations
2. Solve x² − 9 = 0.
Solution
A. x = 3 or x = −3
- x² = 9.
- x = √9 = 3 or x = −3. A quadratic usually has two solutions.
Topic: Quadratic equations
3. Solve 2x² + 3x − 2 = 0.
Solution
D. x = 1/2 or x = −2
- Factorise: (2x − 1)(x + 2) = 0.
- 2x − 1 = 0 gives x = 1/2, and x + 2 = 0 gives x = −2.
Topic: Simultaneous equations
4. Solve 2x + y = 7 and x − y = 2.
Solution
C. x = 3, y = 1
- Add the equations to eliminate y: 3x = 9, so x = 3.
- Substitute: 3 − y = 2, so y = 1.
Topic: Expanding brackets
5. Expand (x + 4)(x − 3).
Solution
B. x² + x − 12
- x × x = x², x × −3 = −3x, 4 × x = 4x, 4 × −3 = −12.
- x² − 3x + 4x − 12 = x² + x − 12.
Topic: Algebraic fractions
6. Simplify (x² − 16)/(x + 4).
Solution
A. x − 4
- x² − 16 is a difference of two squares: (x + 4)(x − 4).
- Cancel (x + 4): x − 4.
Topic: Completing the square
7. Write x² + 6x + 1 in the form (x + a)² + b.
Solution
D. (x + 3)² − 8
- Halve the x coefficient: (x + 3)² = x² + 6x + 9.
- x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8.
Topic: Composite functions
8. f(x) = 3x − 1 and g(x) = x². Find fg(2).
Solution
C. 11
- fg(2) means f(g(2)). Work out g first: g(2) = 4.
- f(4) = 3 × 4 − 1 = 11.
Topic: Inverse functions
9. f(x) = 2x + 5. Find f⁻¹(x).
Solution
B. (x − 5)/2
- Write y = 2x + 5 and make x the subject: x = (y − 5)/2.
- So f⁻¹(x) = (x − 5)/2.
Topic: Quadratic sequences
10. Find the nth term of 2, 5, 10, 17, ...
Solution
A. n² + 1
- Differences: 3, 5, 7. Second differences: 2, so the sequence starts n².
- n² gives 1, 4, 9, 16. Each term is 1 more, so n² + 1.