Question

If x² – 6x + 1 = 0, find the value of `x^4 + 1/x^4`.

Answer 1

Given: x² – 6x + 1 = 0

Step-wise calculation:

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

`x - 6 + 1/x = 0` which rearranges to `x + 1/x = 6`

2. Square both sides:

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

⇒ `x^2 + 2 + 1/x^2 = 36`

3. Simplify to find `x^2 + 1/x^2`:

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

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

4. Square `x^2 + 1/x^2` to find `x^4 + 1/x^4`:

`(x^2 + 1/x^2)^2 = 34^2 = 1156` which expands as `x^4 + 2 + 1/x^4 = 1156`

5. Solve for `x^4 + 1/x^4`:

`x^4 + 1/x^4 = 1156 - 2`

`x^4 + 1/x^4 = 1154`