Page 154 - ICSE Math 8

(x + a)(x + b) = x(x + b) + a(x + b) (Distributive property of multiplication over addition)
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= x + xb + ax + ab = x + bx + ax + ab (Commutative property xb = bx)
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= x + ax + bx + ab ( bx + ax = ax + bx)
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= x + (a + b)x + ab (Taking x common from ax + bx)

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So, you have the identity: \ (x + a)(x + b) = x + (a + b)x + ab
Maths Info
This identity can be used for the following three identities:
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(i) (x + a)(x – b) = x + (a – b)x – ab Some more special products. 3
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1. (a + b) = a + 3ab(a + b) + b
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(ii) (x – a)(x + b) = x + (b – a)x – ab 2. (a – b) = a – 3ab(a – b) – b 3
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(iii) (x – a)(x – b) = x – (a + b)x + ab 3. (a + b + c) = a + b + c + 2(ab + ac + bc)
All these identities are very helpful in simplifying and evaluating algebraic expressions.
Example 10: Find the products: (a) (x + 4)(x + 5) (b) (4x – 3)(4x + 5)
Solution: (a) (x + 4)(x + 5)
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= (x) + (4 + 5)x + 4 ¥ 5 {Using (x + a)(x + b) = x + (a + b)x + ab}
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= x + 9x + 20
(b) (4x – 3)(4x + 5)
\ (4x – 3)(4x + 5) = (y – 3)(y + 5) (Putting 4x = y)
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= y + (–3 + 5)y + (–3) ¥ 5 {Using (x + a)(x + b) = x + (a + b)x + ab}
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= y + 2y – 15 = (4x) + 2(4x) – 15 (Putting y = 4x)
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= 16x + 8x – 15
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Example 11: Evaluate the following using (x + a)(x + b) = x + (a + b)x + ab:
(a) 104 ¥ 101 (b) 54 ¥ 47 (c) 96 ¥ 99
Solution: (a) 104 ¥ 101 = (100 + 4) ¥ (100 + 1) (Here, x = 100, a = 4, b = 1)
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= (100) + (4 + 1)100 + (4 ¥ 1)
= 10,000 + 500 + 4 = 10,504
(b) 54 ¥ 47 = (50 + 4) ¥ (50 – 3) (Here, x = 50, a = 4, b = –3)
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= (50) + {4 + (–3)}50 + 4 ¥ (–3)
= 2,500 + (1)(50) – 12 = 2,500 + 50 – 12 = 2,538
(c) 96 ¥ 99 = (100 – 4) ¥ (100 – 1) (Here, x = 100, a = –4, b = –1)
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= (100) + {(–4) + (–1)}100 + (–4) ¥ (–1)
= 10,000 + (–5)100 + 4 = 10,000 – 500 + 4 = 9,504

EXERCISE 13.2

1. Find the following products:
(a) (x + 2)(x + 6) (b) (x + 3)(x – 4) (c) (x – 7)(x – 5) (d) (4x + 7)(4x – 3)
Ê 3 ˆ Ê 3 ˆ Ê 2 ˆ Ê ˆ 5 Ê 1ˆ Ê 3ˆ
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2
2
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(e) Á 4 y - 2 ˜ Á 4 y + 4 (f) Á x + 5 ˜ Á x + ˜ ¯ 2 (g) (7x – 6xy)(7x – 3xy) (h) Á l - 2¯ Ë l + 4¯ ˜
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˜ Á
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¯
¯ Ë
¯ Ë
Ë
Ë
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2. Evaluate the following using the identity (x + a)(x + b) = x + (a + b)x + ab.
(a) 103 ¥ 105 (b) 108 ¥ 106 (c) 97 ¥ 96 (d) 102 ¥ 95
(e) 59 ¥ 48 (f) 28 ¥ 33 (g) 995 ¥ 1,004 (h) 45 ¥ 47
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