Question

If x² − 6x − 1 = 0, find `x^3-1/x^3`.

Answer 1

x² − 6x − 1 = 0

⇒ x² − 1 = 6x

⇒ `(x^2 - 1)/x = 6`

⇒ `x - 1/x = 6`

Let’s take a = x, b = `1/x`,

Using the identity

a³ − b³ = (a − b)³ + 3ab(a − b)

Thus,

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

`x^3 - 1/x^3 = (6)^3 + 3(6)`

= 216 + 18

∴ `x^3 - 1/x^3 = 234`