(iii) 2(x2 + 1/x2
) – 9(x+1/x) + 14 = 0
Sol. 2(x2 + 1/x2
) – 9(x+1/x) + 14 = 0
∴ 2[(x + 1/x)2
– 2] - 9(x+1/x) + 14 = 0
[∵(x2 + 1/x2 ) = [(x
+ 1/x)2 – 2]
Put x + 1/x
= m,
∴ 2(m2 – 2) – 9m
+ 14 = 0
∴ 2m2 – 4 – 9m + 14
= 0
∴ 2m2 – 9m + 10 =
0
∴ 2m2 – 4m – 5m +
10 = 0
∴ 2m (m – 2) – 5 (m – 2) = 0
∴ (m – 2) (2m – 5) = 0
∴ m – 2 = 0 or 2m – 5 = 0
∴ m = 2 or 2m = 5
∴ m = 2 or m = 5/2
Resubstituting m = x + 1/x
x + 1/x
= 2 ....(i) x + 1/x = 5/2
..... (ii)
From (i)
x + 1/x
= 2
Multiplying by x, we get
x2 + 1 = 2x
∴ x2 – 2x + 1 = 0
∴ x2 – x – x + 1
= 0
∴ x(x – 1) – 1 (x – 1) = 0
∴ (x – 1) (x – 1) = 0
∴ x – 1 = 0 or
x – 1 = 0
∴ x = 1 or
x = 1
i.e. x = 1
From (ii)
x + 1/x
= 5/2
Multiplying by 2x, we get
2x2 + 2 = 5x
∴ 2x2 – 5x + 2 = 0
∴ 2x2 - 4x – 1x +
2 = 0
∴ 2x(x – 2) -1 (x – 2) = 0
∴ (2x – 1)(x – 2) = 0
∴ 2x – 1 = 0 or x – 2 = 0
∴ x = ½ or x = 2
∴ x = 1 or ½ or 2
