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Algebra appears on every GCSE maths paper, often accounting for a quarter of the marks or more. Getting comfortable with the three core skills — expanding brackets, factorising expressions and rearranging formulae — gives you a reliable toolkit for solving equations, proving identities and tackling word problems. This guide walks through each technique with worked examples you can practise straight away.
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Technique One
Expanding means multiplying every term inside a bracket by the term outside it. On a non-calculator paper this is tested directly; on a calculator paper it usually appears inside equations or area problems.
Multiply the outside term by each term inside. Example: 3(2x + 5) = 3 × 2x + 3 × 5 = 6x + 15. Watch for negative signs: −2(4x − 3) = −8x + 6 because −2 × −3 gives +6.
Multiply each term in the first bracket by each term in the second. Example: (x + 3)(x + 2) = x² + 2x + 3x + 6 = x² + 5x + 6. On higher-tier papers you may also meet (2x − 1)(3x + 4) = 6x² + 8x − 3x − 4 = 6x² + 5x − 4.
Write the bracket twice and expand. Example: (x + 4)² = (x + 4)(x + 4) = x² + 4x + 4x + 16 = x² + 8x + 16. A common mistake is writing x² + 16, forgetting the middle term.
Tip: After expanding, always collect like terms and count the number of terms you expect. Two single brackets should give four terms before simplifying; a single bracket with two terms inside gives two terms.
Technique Two
Factorising is the reverse of expanding. You rewrite an expression as a product of brackets or a common factor multiplied by a simpler expression. It is essential for solving quadratic equations and simplifying algebraic fractions.
Look for the highest number and the lowest power of each letter that divides every term. Example: 6x² + 9x = 3x(2x + 3). Always check by expanding your answer mentally.
Find two numbers that multiply to give c and add to give b. Example: x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7. If c is negative, one number is positive and one is negative.
When you see a² − b², write (a + b)(a − b). Example: x² − 25 = (x + 5)(x − 5). This pattern appears frequently in higher-tier simplification and proof questions.
Tip: Before factorising a quadratic, check whether you can first take out a common factor. 2x² + 10x + 12 factorises to 2(x² + 5x + 6) and then to 2(x + 2)(x + 3). Missing the first step loses method marks.
Technique Three
Rearranging means making a different letter the subject. The rule is simple: whatever you do to one side, you must do to the other. Do this in reverse order of operations (undo division before subtraction, and so on).
Example: Make x the subject of y = 3x + 7. Subtract 7 from both sides: y − 7 = 3x. Then divide by 3: x = (y − 7) / 3. Write the final answer with the subject on the left.
Example: Make r the subject of A = πr². Divide by π first: A / π = r². Then square-root both sides: r = √(A / π). Remember the ± if the context allows negative values, though geometry questions usually take the positive root.
Example: Make a the subject of 4a + 5 = 7a − 2b. Collect a terms on one side: 5 + 2b = 3a. Then divide: a = (5 + 2b) / 3. Students often lose marks by dividing before collecting.
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Next Step
Download our free GCSE Maths Algebra Pack with graded questions, mark schemes and model answers. If you would like one-to-one support, our tutors cover every exam board and every tier.