NCERT Solutions for Class 9th Maths Chapter 4 Think and Reflect — Factorisation of Algebraic Expressions Using Identities
Book page 73 Updated on2026-09-08
Q1.
What if we replace b by − b in (a + b)² = a² + 2ab + b²?
Answer
You get the second standard identity, free of charge.
(a + (−b))² = a² + 2a(−b) + (−b)²
↓ a + (−b) = a − b, 2a(−b) = −2ab, (−b)² = +b² (a − b)² = a² − 2ab + b²
Only the middle term changes sign, because it is the only term in which b appears an odd number of times.
Why the substitution is allowed: this is the whole point of the word identity. (a + b)² = a² + 2ab + b² is true for every value of b — so it is in particular true when the number standing in the place of b happens to be −b. Nothing is being assumed; we are simply reading the identity at a different value.
The book also gives the picture. Start with a square of side a and split the side into (a − b) and b:
A square of side a. Cutting off the bottom strip ab and the side rectangle b(a − b) leaves the square (a − b)².
Area of the whole square = a²
Area of the shaded strip along the bottom = ab
Area of the strip on the right = b(a − b)
So (a − b)² = a² − ab − b(a − b)
= a² − ab − ba + b²
= a² − 2ab + b²
Did you know? The two strips overlap in a b × b corner, which is why subtracting ab and b(a − b) — rather than ab and ab — is what puts the +b² back. The picture is doing the same bookkeeping as the algebra.