GCSE Maths Algebra Notes
Algebra is the language of GCSE Maths — manipulating expressions, solving equations and working with graphs. Master it and the rest of the paper gets easier.
Key concepts
- Expression
- A collection of terms with no equals sign, e.g. 3x + 2. You simplify or expand expressions — you do not solve them.
- Equation
- Two expressions joined by an equals sign, e.g. 3x + 2 = 11. You solve equations to find the unknown.
- Like terms
- Terms with exactly the same letters and powers, e.g. 4x and 7x. Only like terms can be added or subtracted.
- Factorising
- Writing an expression as a product of factors, e.g. 6x + 9 = 3(2x + 3). It is the reverse of expanding.
- Always do the same operation to both sides of an equation to keep it balanced.
- Indices add when multiplying (x³ × x² = x⁵) and subtract when dividing (x⁵ ÷ x² = x³).
- A negative times a negative gives a positive — a constant source of sign errors.
Key formulas
Expanding a single bracket
a(b + c) = ab + ac
Difference of two squares
a² − b² = (a + b)(a − b)
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a
Given on the formula sheet for solving ax² + bx + c = 0.
Worked example
Solve 3(x + 4) = 27.
- 1Expand the bracket: 3x + 12 = 27.
- 2Subtract 12 from both sides: 3x = 15.
- 3Divide both sides by 3: x = 5.
Answer: x = 5
Common mistakes
✗ Only multiplying the first term inside a bracket, e.g. 3(x + 4) = 3x + 4.
✓ Multiply the outside number by every term inside: 3x + 12.
✗ Adding unlike terms, e.g. 3x + 2 = 5x.
✓ You can only combine like terms — 3x + 2 stays as 3x + 2.
✗ Losing a minus sign when expanding, e.g. −(x − 3) = −x − 3.
✓ Change the sign of every term: −(x − 3) = −x + 3.
✗ Forgetting to do the operation to both sides of an equation.
✓ Whatever you do to one side, do to the other to keep it balanced.
Quick check
Try these, then reveal the answers.
Simplify 5x + 2x − 3x.Reveal answer
4x
Expand 4(2x − 5).Reveal answer
8x − 20
Solve 2x + 7 = 19.Reveal answer
x = 6
Exam tips
- Command words: 'simplify' means tidy up, 'solve' means find the value, 'expand' means remove brackets, 'factorise' means put brackets back in.
- Show every line of working — method marks are awarded even if your final answer is wrong.
- On a 'solve' question, substitute your answer back in to check it works.
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Frequently asked questions
What is the difference between an expression and an equation?+
An expression has no equals sign and is simplified or expanded; an equation has an equals sign and is solved to find an unknown value.
Is algebra on the calculator or non-calculator GCSE paper?+
Both. Edexcel sits Paper 1 non-calculator and Papers 2 and 3 with a calculator, so practise algebra both ways.
How do I get full marks on algebra questions?+
Write one clear step per line so the examiner can follow your method — method marks are awarded for correct working even when the final answer is wrong.
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