Simplification & Surds · Topic 2 of 8

✨ Algebraic Identities on Numbers

3 exam question types, fully solved

When a question throws huge or symmetric numbers at you — like 326 and 222, or 999² − 998² — it is a signal. The examiner wants you to use an identity, not a calculator.

The identities that do the heavy lifting

• (a+b)² = a² + 2ab + b² and (a−b)² = a² − 2ab + b²

• a² − b² = (a+b)(a−b) — turns a subtraction of squares into an easy product.

• (a+b)² − (a−b)² = 4ab and (a+b)² + (a−b)² = 2(a²+b²)

• a³ + b³ = (a+b)(a² − ab + b²) and a³ − b³ = (a−b)(a² + ab + b²)

The skill is recognising the shape. Once you see it, the messy numbers cancel and you are left with one tiny sum. Example: 999² − 998² = (999+998)(999−998) = 1997 × 1 = 1997 — no squaring at all.

✨Identity ExplorerExpand & verify on real numbers

a²−b² with a = 326, b = 222: (a+b)(a−b) = 548 × 104

a²−b² = 56992

Using a² − b² = (a+b)(a−b): 51² − 49² = (51+49)(51−49) = 100 × 2 = .

Q1The value of 85² − 15² is:

Q2(105)² using (a+b)² with a=100, b=5 is:

Q3If a+b = 9 and ab = 20, then a²+b² is:

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Real exam questions — Identities on Numbers

3 question types · 9 solved examples from real SSC papers

Large symmetric numbers are a signal: use an identity, never raw arithmetic. These shapes turn ugly sums into one line.

How to solve this type

Spot the squared-binomial shapes: (a+b)² = a²+2ab+b² and (a−b)² = a²−2ab+b². When both appear together, use (a+b)²−(a−b)² = 4ab (a minus sign between them) or (a+b)²+(a−b)² = 2(a²+b²) (a plus sign). For two squares subtracted, use a²−b² = (a+b)(a−b). The moment you see large or symmetric numbers, reach for an identity instead of squaring them.

The value of $\dfrac{(326+222)^2-(326-222)^2}{326\times222}$ is:

A3B1C2D4

Simplify $\dfrac{(379+276)^2+(379-276)^2}{379^2+276^2}$ :

A103B655C1D2

If $a+b=5$ and $a-b=1$ , find $a^2+b^2$ and $a^2-b^2$ .

A17 and 5B11 and 5C13 and 5D13 and 10