Chapter 4 · ICSE Class 9 Mathematics (Selina Concise)

Expansions

Master algebraic identities for expanding expressions like (a+b)², (a-b)², and (a+b)(a-b), and apply them to numerical shortcuts.

📐 Chapter 4

The expansion of (a plus b) squared shown as a square split into four areas A square with side a plus b divided into four regions: a square of area a squared, a square of area b squared, and two rectangles of area a times b, showing why (a+b) squared equals a squared plus 2ab plus b squared a² ab ab b² (a+b)² = a²+2ab+b²
4.1

Square Identities

(a+b)²=a²+2ab+b², (a−b)²=a²−2ab+b², and (a+b)(a−b)=a²−b². These identities simplify both algebraic and numerical calculations.

1

Expand (2x+3y)².

4x²+12xy+9y²

2

Find 105×95 using the identity (a+b)(a−b).

(100+5)(100−5)=100²−5²=10000−25=9,975

3

Expand (5x+2y)².

25x²+20xy+4y²

4.2

Finding Products from Sum and Difference

If the sum and difference of two numbers are known, their product and the sum of their squares can be found using expansion identities.

1

If x+y=8 and x−y=2, find xy.

(x+y)²−(x−y)²=4xy → 64−4=4xy → xy=15

2

If x+y=10 and xy=21, find x²+y².

x²+y²=(x+y)²−2xy=100−42=58

3

If x+y=9 and xy=14, find x²+y².

x²+y²=(x+y)²−2xy=81−28=53

Practice

Exercises

1. Expand (3a−4b)².

2. Find 998×1002 using an identity.

3. If x+y=6 and x−y=2, find xy.

4. If x−y=4 and xy=12, find x²+y².

5. Expand (x+2y)(x−2y).

6. Expand (2a-3b)².

7. Find 997×1003 using an identity.

8. If x+y=10 and x-y=4, find xy.

9. If x-y=5 and xy=24, find x²+y².

10. Expand (2x+5y)(2x-5y).

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