Question

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

Answer 1

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

We need to find the value of `x - 1/x`

Step-wise calculation:

1. First, find `1/x`:

\[ \frac{1}{x} = \frac{1}{5 - 2\sqrt{6}} \]

Rationalize the denominator:

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

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

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

2. Now calculate `x - 1/x`:

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

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

`x - 1/x = -4sqrt(6)`