NCERT Solutions for Class 9th Maths Chapter 4 Think and Reflect — Visualising Identities

Book page 71 Updated on2026-09-08

Q1.

What can you say about a and b if (a + b)² < a² + b²?

Answer

Everything is decided by one term. Subtract:

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

So (a + b)² < a² + b² exactly when 2ab < 0, that is when

ab < 0 — a and b have opposite signs (one positive, one negative, neither zero)

Check: a = 10, b = −2 gives (a + b)² = 8² = 64 while a² + b² = 100 + 4 = 104, and indeed 64 < 104.

Why it happens: The two expressions differ by the cross term 2ab and by nothing else. Both a² and b² are never negative, so they can never decide the comparison — only the sign of the product ab can. This is exactly the hint the book gives: use the expansion to find the deciding term.

Q2.

What can you say about a and b if (a + b)² > a² + b²?

Answer

Same difference, opposite sign.

(a + b)² − (a² + b²) = 2ab > 0 ⇒ ab > 0
⇒ a and b have the same sign — both positive, or both negative

Check both cases: a = 10, b = 2 gives 144 > 104. And a = −2, b = −3 gives (−5)² = 25 while a² + b² = 4 + 9 = 13, so 25 > 13.

Tip: The book’s example used two positive numbers, which is why (a + b)² came out larger. Two negative numbers behave the same way, because their product is still positive.

Q3.

When will (a + b)² be equal to a² + b²?

Answer

(a + b)² = a² + b² ⇔ 2ab = 0 ⇔ ab = 0
⇔ a = 0 or b = 0 (or both)

For instance a = 7, b = 0: (7 + 0)² = 49 and 7² + 0² = 49.

Why it happens: A product of two real numbers is zero only when at least one factor is zero — there are no other possibilities. So the equation (a + b)² = a² + b² is not an identity: it is an ordinary equation, satisfied only on the two lines a = 0 and b = 0. This is precisely the distinction the section is making. An identity such as (a + b)² = a² + 2ab + b² holds everywhere; an equation such as (a + b)² = a² + b² holds only at special values.

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