Question

If `x = 5 - 2sqrt(6)`, find the value of `x^2 + 1/x^2`.

Answer 1

Given: `x = 5 - 2sqrt(6)`

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

Step-wise calculation:

1. First, find `1/x`:

`1/x = 1/(5 - 2sqrt(6))`

Rationalize the denominator:

`1/x = (5 + 2sqrt(6))/((5 - 2sqrt(6))(5 + 2sqrt(6))`

`1/x = (5 + 2sqrt(6))/(25 - 24)`

`1/x = 5 + 2sqrt(6)`

2. Calculate `x + 1/x`:

`x + 1/x = (5 - 2sqrt(6)) + (5 + 2sqrt(6))`

`x + 1/x = 10`

3. Find `x^2 + 1/x^2` using the identity:

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

Substituting,

`10^2 = x^2 + 2 + 1/x^2`

⇒ `100 = x^2 + 1/x^2 + 2`

Therefore,

`x^2 + 1/x^2 = 100 - 2`

`x^2 + 1/x^2 = 98`

So, the value of `x^2 + 1/x^2` when `x = 5 - 2sqrt(6)` is 98.