Q1.
Why is this identity true? [a² = (a + b) (a – b) + b²]
Answer
It is Identity 1C rearranged. Start from
(a + b)(a – b) = a² – b²
and add b² to both sides:
(a + b)(a – b) + b² = a² – b² + b²
a² = (a + b)(a – b) + b²
a² = (a + b)(a – b) + b²
This turns a squaring into a multiplication of two numbers that are usually much friendlier.
| Square | Choice of b | Working | Answer |
|---|---|---|---|
| 31² | b = 1 | 32 × 30 + 1 | 961 |
| 197² | b = 3 | 200 × 194 + 9 | 38809 |
| 48² | b = 2 | 50 × 46 + 4 | 2304 |
Why it happens: choose b so that a + b (or a – b) becomes a multiple of 10. The awkward square is then traded for an easy product plus a tiny square. Sridharacharya (750 CE) used exactly this.