GCSE Maths Algebraic manipulation revision

Algebra is shorthand with strict rules. 3y means 3 × y, a²b means a × a × b, and 3a² means 3 × a × a — the power acts on the letter, not on the coefficient. A number in front is a coefficient (in ¾x the coefficient is ¾, written as a fraction rather than a decimal). Terms are the building blocks separated by + or −; factors are the pieces multiplied together.

Classify statements carefully: 2x + 3 is an expression (no equals sign); 2x + 3 = 9 is an equation (true for particular values); A = ½bh is a formula (variables from the real world); 2(x + 3) ≡ 2x + 6 is an identity (true for every value). Spotting the difference matters — “simplify” belongs to expressions and identities, “solve” belongs to equations.

Like terms share exactly the same letter part: 5a and 3a collect to 8a, but 5a and 3a² never collect — they are different shapes. Simplify 4x + 3y − x + 5y by grouping: (4 − 1)x + (3 + 5)y = 3x + 8y. Keep the sign attached to its term when you regroup: 7p − 2q − 10p + q = −3p − q.

A bracket spreads its multiplier over every term: 5(2x − 3) = 10x − 15, and 4x(3x + 1) = 12x² + 4x. A negative multiplier flips signs: −3(4 − x) = −12 + 3x, which is tidier written as 3x − 12. The most common error is multiplying the first term only: −2(x − 5) = −2x + 10, not −2x − 10.

To expand (x + 2)(x + 3), let each term in the first bracket meet each term in the second: x² + 3x + 2x + 6 = x² + 5x + 6. The grid and the “fish” both do the same job — the middle coefficient is the sum of the bracket numbers and the end constant is their product. With coefficients: (2x − 1)(x + 5) = 2x² + 10x − x − 5 = 2x² + 9x − 5.

Three brackets expand pairwise first. Squared brackets need care: (x + 4)² = (x + 4)(x + 4) = x² + 8x + 16 — never x² + 16. Watch negatives: (x − 3)(x − 2) = x² − 5x + 6, where two minus multiplications produce the +6.

Common factors come out first, always: 4x² + 8x = 4x(x + 2) and 6x²y − 9xy² = 3xy(2x − 3y). Take out the highest common factor of the numbers and all shared letters. Then, for x² + bx + c, find two numbers with product c and sum b: x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7; x² − 5x − 14 = (x − 7)(x + 2) because −7 × 2 = −14 and −7 + 2 = −5.

The difference of two squares is the pattern to spot on sight: a² − b² = (a + b)(a − b). So x² − 9 = (x + 3)(x − 3), and 4x² − 25 = (2x + 5)(2x − 5) — square-root the ends. There is no middle term to find; if you are expanding (x + 3)(x − 3) and see a middle appear, something slipped.

Higher. Simplify an algebraic fraction by factorising everything, then cancelling common factors: (x² − 9) ÷ (x + 3) becomes (x + 3)(x − 3)⁄(x + 3) = x − 3. You may cancel a bracket that divides the whole of the top and bottom — never a single term: (x² + 9)⁄(x + 3) does not simplify, and cancelling the x’s there is illegal.

Adding and subtracting needs a common denominator, exactly like numeric fractions: (x + 2)⁄3 + (x − 1)⁄4 = (4(x + 2) + 3(x − 1))⁄12 = (4x + 8 + 3x − 3)⁄12 = (7x + 5)⁄12. Multiplying fractions multiplies across, after factorising and cancelling first; dividing flips the second fraction.

Higher. The index laws transfer straight to letters: x⁵ × x³ = x⁸, x⁸ ÷ x³ = x⁵, (2x²)³ = 2³x⁶ = 8x⁶, and 12a⁴ ÷ 3a² = 4a². Numbers multiply separately from letters, and negative or fractional indices work exactly as in Chapter 3: 16x⁴ raised to the ½ is 4x², and x⁻³ = 1⁄x³.

Surds simplify the same way: √(18x²) = 3x√2 (for positive x), and bracket expansions can mix letters and surds: (√5 + 2)(√5 − 3) = 5 − 3√5 + 2√5 − 6 = −1 − √5. Collect like surds at the end, exactly like like terms.