NCERT Solutions Ganita Prakash (Part 1) Chapter 6 .2 Special Cases of the Distributive Property — In-text Questions
Book page 1476 Updated on2026-09-05
Q1.
We have seen what (a + b)² gives when expanded. What is the expansion of (a – b)²?
Answer
Multiply the bracket by itself, using the distributive property.
(a – b)² = (a – b) × (a – b) = a² – ba – ab + b² = a² – 2ab + b² Identity 1B: (a – b)² = a² + b² – 2ab
Test at a = 9, b = 4: LHS = 5² = 25; RHS = 81 + 16 – 72 = 25 ✓
Why it happens: the four products are a·a, a·(–b), (–b)·a and (–b)(–b). Two of them are –ab, giving –2ab; the last is +b² because a negative times a negative is positive. This is the algebraic twin of the picture on page 146: subtract two strips, add the corner back.
Q2.
We can also use the expansion of (a + b)² to find the expansion of (a – b)². Think how. Hint: (a – b)² = (a + (–b))².
Answer
A subtraction is an addition of the opposite, so Identity 1A already covers it.
(a – b)² = (a + (–b))² = a² + (–b)² + 2 × a × (–b) = a² + b² – 2ab
The only two things used are (–b)² = b² and a × (–b) = –ab.
Why it happens: Identity 1A was proved for any numbers a and b, so we are free to put –b in place of b. Getting a second identity out of the first by substitution — rather than by starting again — is one of the real economies algebra offers.
Q3.
Find the general expansion of (a – b)² using geometry, as we did for 55².
Answer
Draw a square of side a. Mark off a square of side (a – b) inside it, in one corner.
Area of the big square = a² Remove the strip along the right: a × b Remove the strip along the bottom: b × a The corner b × b has now gone twice, so add it back: (a – b)² = a² – ab – ba + b² = a² – 2ab + b²
The square of side a splits into (a – b)², two strips of b(a – b) and the corner b². Since 2b(a – b) + b² = 2ab – b², the shaded square is a² – 2ab + b².
Reading the picture the other way round: a² = (a – b)² + 2b(a – b) + b², and expanding 2b(a – b) = 2ab – 2b² gives (a – b)² = a² – 2ab + b² again.
Q4.
Use the identity (a – b)² to find the values of (a) 99² and (b) 58².
Tip: the split to look for is “a round number, minus a small number”. Squaring 99 or 58 straight out takes a full multiplication; this way it is two easy squares and one doubling.
Q5.
Expand the following using both Identity 1B and by applying the distributive property (i) (b – 6)² (ii) (–2a + 3)² (iii) (7y – 3⁄4 z)²