Question

If x² – 6x + 1 = 0, find the value of `x^3 + 1/x^3`.

Answer 1

Given: x² – 6x + 1 = 0

Step-wise calculation:

1. Divide the equation by (x) assuming (x ≠ 0):

`x - 6 + 1/x = 0`

⇒ `x + 1/x = 6`

2. Square both sides to find `x^2 + 1/x^2`:

`(x + 1/x)^2 = 6^2`

⇒ `x^2 + 2 + 1/x^2 = 36`

`x^2 + 1/x^2 = 36 - 2`

`x^2 + 1/x^2 = 34`

3. Use the identity to find `x^3 + 1/x^3`:

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

Substitute `x + 1/x = 6`:

`6^3 = x^3 + 1/x^3 + 3 xx 6`

`216 = x^3 + 1/x^3 + 18`

`x^3 + 1/x^3 = 216 - 18`

`x^3 + 1/x^3 = 198`