Question

If `x^2 + 1/x^2` = 23, then find the value of `x + 1/x` and `x^3 + 1/x^3`

Answer 1

`x^2 + 1/x^2` = 23

`(x + 1/x)^2 - 2` = 23  ...[a² + b² = (a + b)² − 2ab]

`(x + 1/x)^2` = 23 + 2

⇒ `(x + 1/x)^2` = 25

 `x + 1/x = sqrt(25)`

`x+  1/x` = ± 5

`x^3 + 1/x^3 = (x + 1/x)^3 - 3x xx 1/x(x + 1/x)`

When x = 5 ...[a³ + b³ = (a + b)³ – 3ab(a + b)]

= (5)³ – 3(5)

= 125 – 15

= 110

when x = – 5

`x^3 + 1/x^3` = (–5)³ – 3(–5)

= – 125 + 15

= – 110

∴ `x^3 + 1/x^3` = ± 110