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.

---

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

---

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.