Algebra: Expressions and Quadratics

At GCSE you extend algebraic manipulation to include triple-bracket expansion, all forms of factorisation, algebraic fractions and completing the square.

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Expanding Brackets

Double Brackets (Review)

(2x + 3)(x − 5) = 2x² − 10x + 3x − 15 = 2x² − 7x − 15

Triple Brackets

Expand two brackets first, then multiply the result by the third.

Example: (x + 2)(x − 1)(x + 3) Step 1: (x + 2)(x − 1) = x² + x − 2 Step 2: (x² + x − 2)(x + 3) = x³ + 3x² + x² + 3x − 2x − 6 = x³ + 4x² + x − 6

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Factorising

Taking Out a Common Factor

12x³y − 8x²y² = 4x²y(3x − 2y)

Difference of Two Squares

a² − b² = (a + b)(a − b)

9x² − 25 = (3x + 5)(3x − 5) 16 − x² = (4 + x)(4 − x)

Factorising Quadratics x² + bx + c

Find two numbers that multiply to c and add to b.

x² − x − 12: numbers → ×(−12) and +(−1) → −4 and 3 → (x − 4)(x + 3)

Factorising ax² + bx + c (a ≠ 1)

Method (grouping/splitting the middle term): 6x² + 7x − 3: a × c = −18; find two numbers → ×(−18) and +7 → 9 and −2 6x² + 9x − 2x − 3 = 3x(2x + 3) − 1(2x + 3) = (3x − 1)(2x + 3)

Factorising by Grouping (four terms)

ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

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Algebraic Fractions

Simplifying

Cancel common factors (factorise numerator and denominator first).

(x² + 5x + 6)/(x² − 4) = (x + 2)(x + 3)/((x + 2)(x − 2)) = (x + 3)/(x − 2)

Adding and Subtracting

Find a common denominator.

3/(x + 1) + 2/(x − 2) = 3(x − 2)/((x+1)(x−2)) + 2(x + 1)/((x+1)(x−2)) = (3x − 6 + 2x + 2)/((x+1)(x−2)) = (5x − 4)/((x+1)(x−2))

Multiplying and Dividing

Multiply: (a/b) × (c/d) = ac/bd — factorise and cancel first. Divide: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc

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Completing the Square