Algebra is the spine of GCSE Maths. Almost half of every Higher paper, and a third of every Foundation paper, sits on top of algebraic skills — even questions about graphs, statistics or geometry usually need an algebraic step somewhere. Get the four core skills below sharp and the entire paper gets easier. This guide walks through expanding, factorising, solving and rearranging, plus the small handful of rules students keep forgetting and that examiners keep penalising.
The golden rule
Whatever you do to one side of an equation, you do to the other. Whatever you do to the top of a fraction, you do to the bottom (within reason). Algebra is consistent — break the consistency, lose the mark.
Skill 1 — Expanding brackets
To expand brackets, multiply every term inside by the term outside. With two brackets, multiply every term in the first by every term in the second.
Worked example 1 — single bracket
Expand 3(2x + 5).
3 × 2x + 3 × 5 = 6x + 15
Worked example 2 — double bracket (FOIL)
Expand (x + 3)(x + 7).
Multiply each term in the first bracket by each in the second — F O I L: First, Outer, Inner, Last.
x × x + x × 7 + 3 × x + 3 × 7
= x² + 7x + 3x + 21 = x² + 10x + 21
−(x − 5) is −x + 5, not −x − 5. Sign errors here are the single most common reason students lose easy marks.
Skill 2 — Factorising
Factorising is expanding in reverse — taking a polynomial and writing it as a product. There are four flavours at GCSE.
Common factor
Worked example 3
Factorise 6x² + 9x.
Both terms share 3x.
3x(2x + 3)
Difference of two squares
Pattern: a² − b² = (a + b)(a − b). Watch for one square term minus another.
Worked example 4
Factorise x² − 49.
x² − 7² = (x + 7)(x − 7)
Quadratic in the form x² + bx + c
Find two numbers that multiply to give c and add to give b. (Covered in detail in the quadratic equations guide.)
Skill 3 — Solving linear equations
Solving means finding the value of the unknown. The strategy is to undo the operations one at a time, keeping the equation balanced.
Worked example 5
Solve 3x + 7 = 22.
3x = 22 − 7 = 15
x = 15 ÷ 3 = 5
Worked example 6 — unknown on both sides
Solve 5x − 4 = 2x + 11.
Get all the x's on one side and the numbers on the other.
5x − 2x = 11 + 4
3x = 15
x = 5
Skill 4 — Rearranging formulae
Rearranging is the same skill as solving, but with letters instead of numbers. Treat the letter you want to isolate exactly as you'd treat x.
Worked example 7
Make r the subject of C = 2πr.
C = 2πr
C / (2π) = r
So r = C / (2π).
Worked example 8 — Higher tier
Make x the subject of y = (x + 3) / (x − 1).
Multiply both sides by (x − 1):
y(x − 1) = x + 3
yx − y = x + 3
Bring x's together:
yx − x = y + 3
x(y − 1) = y + 3
x = (y + 3) / (y − 1)
The mistakes that cost the most marks
−(x − 5) = −x + 5. Always.
3(x + 4) is 3x + 12, not 3x + 4. Multiply the 3 by everything inside.
(x² + 3x) / x, you can cancel only because x is a factor of both terms (the answer is x + 3). You can never cancel (x + 5) / x to give 5.
Where this leads next
Once these four skills are solid, the rest of the algebra syllabus follows naturally: quadratic equations (factorising and the formula), algebraic fractions (cancelling factors, adding fractions), and algebraic proof (using algebra to show that a statement is always true).
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