Question

If \[x = \sqrt{6} + \sqrt{5}\],then \[x^2 + \frac{1}{x^2} - 2 =\]

Options

• \[2\sqrt{6}\]

• \[2\sqrt{5}\]

• 24

• 20

Answer 1

Given that `x = sqrt6 +sqrt5 ` .Hence `1/x`is given as

`1/x = 1/(sqrt6+sqrt5)`.We need to find  `x^2 +1/x^2 - 2`

We know that rationalization factor for  `sqrt6 +sqrt5` is`sqrt6 -sqrt5`. We will multiply numerator and denominator of the given expression   `1/(sqrt6+sqrt5)`by `sqrt6 -sqrt5`, to get

`1/x = 1/(sqrt6+sqrt5) xx (sqrt6-sqrt5)/(sqrt6-sqrt5) `

` = (sqrt6-sqrt5)/((sqrt6)^2 - (sqrt5)^2)`

` = (sqrt6  - sqrt5)/(6-5)`

` = sqrt6 - sqrt5.`

We know that `(x-1/x)^2 = x^2 + 1/x^2 - 2 ` therefore,

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

` = (sqrt 6 + sqrt5 - (sqrt6  - sqrt5))^2`

` = (sqrt6 + sqrt5 - sqrt6 +sqrt5)^2`

` = (2sqrt5)^2`

`= 20`