Page 151 - ICSE Math 8

Ê 1ˆ Ê 1ˆ 2
2
2
2
2
fi x + 2(x) Á ˜ + Á ˜ = 4 {Using (a + b) = a + 2ab + b }
Ë x¯ Ë x¯
1 1
2
2
fi x + 2 + = 4 fi x + = 4 – 2 = 2 (Transposing 2 to the RHS)
x 2 x 2
1
2
(b) x + = 2 {From part (a)}
x 2
Ê 1 ˆ 2
2
2
fi x + x ¯ = 2 () (Squaring both sides)
2 ˜
Á
Ë
2 Ê
1 ˆ Ê
1 ˆ
2 2
2
2
2
+
fi (x ) + 2(x ) Á x ¯ Ë x ¯ 2 = 4 {Using (a + b) = a + 2ab + b }
2 ˜
2 ˜ Á
Ë
4
4
fi x + 2 + 1 = 4 fi x + 1 = 4 – 2 = 2 (Transposing 2 to the RHS)
x 4 x 4
1 1 1
2
Example 3: If x + = 4 and x + = 10, find the value of x – .
x x 2 x
1
Solution: x + = 4
x
1
Ê ˆ 1 2 Ê 1ˆ Ê ˆ 2 1
2
2
2
2
2
+
Á x + x¯ ˜ = x + 2(x) Á ˜ Á ˜ = x + 2 + x 2 ...(1) {Using (a + b) = a + 2ab + b }
Ë
Ë
x¯ Ë ¯
x
1
Ê ˆ 1 2 Ê 1ˆ Ê ˆ 2 1
2
2
2
2
2
+
and x - x¯ = x – 2(x) Á ˜ Á ˜ = x – 2 + x 2 …(2) {Using (a – b) = a – 2ab + b }
˜
Á
Ë
x¯ Ë ¯
Ë
x
Adding (1) and (2), we get
Ê ˆ 1 2 Ê ˆ 1 2 1 1 2 Ê 1 ˆ
2
2
2
2
Á x + x¯ ˜ + Á x - x¯ = x + +2 x 2 + x -+2 x 2 = 2x + x 2 = 2 x + x ¯
Á
˜
2 ˜
Ë
Ë
Ë
1 1
2
Putting the values of x + and x + , we get
x x 2 Maths Info
2
Ê
2
(4) + x - x¯ ˆ 1 ˜ = 2 ¥ 10 Some common mistakes:
Á
Ë
Ê ˆ 1 2 1. ax is misinterpreted as (a + x). ax means
(a × x) not (a + x).
fi 16 + x - x¯ ˜ = 20 2. Wrong application of distributive
Á
Ë
property:
Ê ˆ 1 2 a(b + c) = (a × b) + (a × c) not ab + c
Á x - x¯ ˜ = 20 – 16 = 4 = (2) 2 3. (a ) = a , not a m + n
Ë
mn
m n
m m
m
1 4. (ab) = a b , not ab m
fi x – = ± 2
x
2
2
Example 4: If 2x + 3y = 10 and xy = 5, find the value of 4x + 9y .
Solution: 2x + 3y = 10
2
2
fi (2x + 3y) = (10) (Squaring both sides)
2
2
2
2
2
fi (2x) + 2(2x)(3y) + (3y) = 100 {Using (a + b) = a + 2ab + b }
2
2
2
2
fi 4x + 12xy + 9y = 100 fi (4x + 9y ) + 12xy = 100
2
2
2
2
fi (4x + 9y ) + 12(5) = 100 fi 4x + 9y + 60 = 100 (Putting the value of xy)
2
2
fi 4x + 9y = 100 – 60 = 40
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