Expanding Brackets: Key Algebraic Identities for GCSE and A-Level

This page collects the most important algebraic expansions involving brackets. All expressions use real numbers or variables such as a, b, c, x.

1. Squares of Binomials

Expression Expanded form
(a + b)2 a2 + 2ab + b2
(a - b)2 a2 - 2ab + b2
(x + a)2 x2 + 2ax + a2
(x - a)2 x2 - 2ax + a2

2. Product of a Sum and a Difference

Expression Expanded form
(a + b)(a - b) a2 - b2
(ax + b)(ax - b) a2x2 - b2

3. Quadratics from Two Brackets

Expression Expanded form
(x + a)(x + b) x2 + (a + b)x + ab
(x - a)(x - b) x2 - (a + b)x + ab
(x + a)(x - b) x2 + (a - b)x - ab
(px + q)(rx + s) pr x2 + (ps + qr)x + qs

4. Cubes of Binomials

Expression Expanded form
(a + b)3 a3 + 3a2b + 3ab2 + b3
(a - b)3 a3 - 3a2b + 3ab2 - b3
(x + a)3 x3 + 3x2a + 3xa2 + a3
(x - a)3 x3 - 3x2a + 3xa2 - a3

5. Squares of Three-Term Expressions

Expression Expanded form
(a + b + c)2 a2 + b2 + c2 + 2ab + 2ac + 2bc
(x + a + b)2 x2 + a2 + b2 + 2ax + 2bx + 2ab

6. Special Cube Identities (Related to Brackets)

These are often used together with bracket expansions:

Expression Expanded form
a3 + b3 (a + b)(a2 - ab + b2)
a3 - b3 (a - b)(a2 + ab + b2)

These formulas cover the most commonly used bracket expansions in algebra, especially for GCSE and A-Level work.

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