NCERT Solutions for Class 9th Maths Chapter 4 In-text Questions — More Identities
Book page 77 Updated on2026-09-08
Q1.
Look at the following figure (Fig. 4.5). Justify the identity a² = (a + b)(a − b) + b² for yourself.
Answer
The figure is a cut-and-slide proof. Start with a square of side a and take a strip of width b off the bottom.
A square of side a. The bottom strip is cut at width b: the piece b(a − b) swings round to the right-hand side, turning the shape into a rectangle (a + b) by (a − b), and the b × b corner is left over.
Before the cut: area = a²
The bottom strip is a × b, and it splits into b(a − b) and b².
Slide the piece b(a − b) round to the right-hand edge. It fits exactly, because the remaining block is (a − b) tall.
After the slide: a rectangle of width a + b and height a − b, plus a leftover square b²
area = (a + b)(a − b) + b²
Nothing was added or thrown away, so a² = (a + b)(a − b) + b²
Rearranged, this is the identity you met in Grade 8:
a² − b² = (a + b)(a − b)
Why the pieces fit: the block left after removing the bottom strip is a wide and (a − b) tall. The strip we cut off is b tall and (a − b) long, so standing it on end makes it (a − b) tall and b wide — exactly the height of the block. Widths add: a + b. Heights match: a − b. The identity is a statement that a rearrangement is possible, and the picture is the rearrangement.
Did you know? In 750 CE Śhrīdharāchārya proposed this as a way to square numbers mentally. Choose b so that a − b and a + b are easy: 55² = (55 + 5)(55 − 5) + 5² = 60 × 50 + 25 = 3025. Try 98² = 100 × 96 + 4 = 9604, and 43² = 46 × 40 + 9 = 1849.