Identities and algebraic expansion

2026-10-10Algebraic Manipulation Ms Chu

Use standard identities to expand quickly and to recognise structure in reverse for factorisation.

Key formulas

(a+b)^{2} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2} - 2ab + b^{2}

(a+b)(a-b) = a^{2} - b^{2}

Identity versus equation

An identity is true for all admissible values of the variables. The three square and difference-of-squares identities should be memorised cold — they appear in almost every Secondary 3 algebra paper.

Expanding with identities

Match the pattern before expanding term by term. For (2x + 3)^2, take a = 2x and b = 3 to get 4x² + 12x + 9. Check the middle term: 2ab must use the actual a and b, including coefficients.

Difference of squares

(a + b)(a − b) expands to a² − b². Spotting this pattern saves time and prepares factorisation.

Exam habit

After expanding, mentally substitute a simple number (e.g. x = 1) to check both sides roughly.

Worked example

Square identity

Problem. Expand (5x − 2)^2.

Solution. a = 5x, b = 2. a² − 2ab + b² = 25x² − 20x + 4.

Review quiz

1 / 4

(x+5)^2 =

Practice questions

Expand (x + 5)^2 and (x − 5)^2.
Expand (3x + 2)(3x − 2).
Expand (2x − 7)^2.

Study notes

Is (a+b)^2 equal to a^2 + b^2?

No — the middle term 2ab is required.

Do identities need proving every time?

You may quote standard identities; expand from first principles if asked to prove.