Question

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Answer 1

Answer :

(i) x² + 1/x² = 2

(ii) x⁴ + 1/x⁴ = 2

Solution :

• Given : x + 1/x = 2 , x ≠ 0
• To find : (i) x² + 1/x² (ii) x⁴ + 1/x⁴

(i) We have ,

x + 1/x = 2

Now ,

Squaring both the sides , we have ;

=> (x + 1/x)² = 2²

=> x² + (1/x)² + 2•x•(1/x) = 4

=> x² + 1/x² + 2 = 4

=> x² + 1/x² = 4 - 2

=> x² + 1/x² = 2

(ii) Now ,

Again squaring both the sides , we have ;

=> (x² + 1/x²)² = 2²

=> (x²)² + (1/x²)² + 2•x²•(1/x²) = 4

=> x⁴ + 1/x⁴ + 2 = 4

=> x⁴ + 1/x⁴ = 4 - 2

=> x⁴ + 1/x⁴ = 2

Hence ,

x² + 1/x² = 2

x⁴ + 1/x⁴ = 2

★ Note :

If x + 1/x = 2 , then

=> (x² + 1)/x = 2

=> x² + 1 = 2x

=> x² - 2x + 1 = 0

=> (x - 1)² = 0

=> x - 1 = 0

=> x = 1

Thus ,

xⁿ + 1/xⁿ = 1