Question

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

Answer 1

Given: `x = 2 + sqrt(3)`

We want to find: `x^2 + 1/x^2`

Stepwise calculation:

1. First, find `1/x`:

`1/x = 1/(2 + sqrt(3))`   ...(By rationalizing the denominator)

`1/x = 2 - sqrt(3)`  

2. Calculate `(x + 1/x)`:

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

`x + 1/x = 4`

3. Use the identity:

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

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

⇒ `16 = x^2 + 1/x^2 + 2`

4. Therefore, `x^2 + 1/x^2`

= 16 – 2 

= 14