If x – 1/x = √5, find the values of

If x – 1/x = √5, find the values of

(i) x² + 1/x²

(ii) x + 1/x

(iii) x³ + 1/x³

Answer :

(i) x² + 1/x² = (x – 1/x)² + 2

Substituting the values

= (√5)² + 2

= 5 + 2

= 7

(ii) (x + 1/x)² = x² + 1/x² + 2

Substituting the values

= 7 + 2

= 9

Here

(x + 1/x)² = 9

So we get

(x + 1/x) = ± √9 = ± 3

(iii) x³ + 1/x³ = (x + 1/x)³ – 3x (1/x) (x + 1/x)

Substituting the values

= (± 3)³ – 3 (± 3)

By further calculation

= (± 27) – (± 9)

= ± 18

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