Question

If `x = 3 + 2sqrt(2)`, find the value of `x^2 + 1/x^2`.

Answer 1

Given: `x = 3 + 2sqrt(2)`

Find: `x^2 + 1/x^2`

Step-wise calculation:

1. First, find the reciprocal `1/x`:

`1/x = 1/(3 + 2sqrt(2))`

`1/x = (3 - 2sqrt(2))/((3)^2 - (2sqrt(2))^2`

`1/x = (3 - 2sqrt(2))/(9 - 8)`

`1/x = 3 - 2sqrt(2)`

2. Find `x + 1/x`:

`x + 1/x = (3 + 2sqrt(2)) + (3 - 2sqrt(2))`

`x + 1/x = 6`

3. Use the identity:

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

4. Substitute the value:

`x^2 + 1/x^2 = 6^2 - 2`

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

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