Question

If `x = sqrt(2) + 1`, find `x^2 + 1/x^2`

Answer 1

Given: `x = sqrt(2) + 1`

Stepwise calculation:

1. Find `1/x`:

`1/x = 1/(sqrt(2) + 1)`

= `(sqrt(2) - 1)/((sqrt(2) + 1)(sqrt(2) - 1))`

= `sqrt(2) - 1`

2. Calculate `x + 1/x`: 

`x + 1/x = (sqrt(2) + 1) + (sqrt(2) - 1)`

`x + 1/x = 2sqrt(2)`

3. Use the identity:

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

⇒ `x^2 + 1/x^2 = (x + 1/x)^2 - 2`

Substitute `(x + 1/x = 2sqrt(2)):`

`x^2 + 1/x^2 = (2sqrt(2))^2 - 2`

= 8 – 2

= 6