Chapter 3 · Expansion & Factorisation

1.1.8 Algebraic Identities

Three shapes are worth knowing by heart — once you do, squaring a number like 1998 × 2002 in your head stops being a trick and starts being obvious.

🎯 By the end of this you can…

✓Expand using (a+b)², (a−b)² and a²−b² without multiplying everything out longhand
✓Evaluate large squares and products mentally using these identities
✓Use the identities on word problems built from x+y, x−y and xy

🟧 See why — the area model for (a + b)²

A square of side $a + b$ splits into four tiles: a square of side $a$ , a square of side $b$ , and two rectangles of $a \times b$ each.

💡 The idea

Three shapes worth memorising

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

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

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

The first two are called perfect squares; the third is the difference of two squares.

✏️ Worked example

Expand (3b + 2)²

1
Match to $a = 3 b$ , $b = 2$ : $(a + b)^{2} = a^{2} + 2 ab + b^{2}$
2
$(3 b)^{2} + 2 (3 b) (2) + 2^{2} = 9 b^{2} + 12 b + 4$

✅ Answer: $9 b^{2} + 12 b + 4$

✏️ Worked example

Expand (s − 5)(s + 5)

1
This matches $(a + b) (a - b) = a^{2} - b^{2}$ with $a = s$ , $b = 5$

✅ Answer: $s^{2} - 25$

🧮 1998 × 2002, without a calculator

Rewrite both numbers around a round number:

1998 = 2000 - 2

and

2002 = 2000 + 2

1 / 4

✏️ Worked example

497², without a calculator

1
Rewrite $497 = 500 - 3$ , so match $a = 500, b = 3$ in $(a - b)^{2} = a^{2} - 2 ab + b^{2}$ .
2
$50 0^{2} - 2 (500) (3) + 3^{2} = 250000 - 3000 + 9$

✅ Answer: $247009$

⚠️ The classic trap

In $(a - b)^{2}$ , the middle term is always subtracted: $a^{2} - 2 ab + b^{2}$ . Adding it by mistake is the single most common slip with this identity — always double-check the sign between the three terms matches the bracket you started with.

✏️ Worked example

If x² + y² = 80 and xy = 12, find (x − y)²

1
Rearrange the identity: $(x - y)^{2} = x^{2} + y^{2} - 2 x y$
2
$= 80 - 2 (12) = 80 - 24$

✅ Answer: 56

✏️ Worked example

If x + y = 10 and x − y = 4, find x² − y²

1
$x^{2} - y^{2} = (x + y) (x - y)$
2
$= (10) (4)$

✅ Answer: 40

🎮 Expand, evaluate, or apply the identity

⭐ 0 XP🔥 streak 0

Expand

(c + 4 d)^{2}

Expand

(5 n - 4)^{2}

Expand

(2 t + 11) (2 t - 11)

Evaluate 102² without a calculator

Evaluate 497² without a calculator

x^{2} + y^{2} = 80

and

x y = 12

, find

(x - y)^{2}

Expand

(a - 6)^{2}

Evaluate 1998 × 2002 using the difference of squares

0 of 8 finished

🤔 Which identity directly expands (w − 10x)(w + 10x)?

🏅

You finished this lesson!

Three identities now replace a page of longhand multiplication — and with them, even a number like 1998 × 2002 is a two-line calculation.

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