Question

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

Options

• \[2\sqrt{5}\]

• 4

• 2

• \[\sqrt{5}\]

Answer 1

Given that. `x=sqrt5 +2 ` Hence  `1/x`is given as

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

We know that rationalization factor for `sqrt5+2` is`sqrt5-2`. We will multiply numerator and denominator of the given expression\`1/(sqrt5 +2)` by`sqrt5 - 2`, to get

`1/x = 1/(sqrt5+2 ) xx (sqrt5 - 2)/(sqrt5 -2)`

 ` = (sqrt 5-2)/((sqrt5)^2 - (2)^2 )`

`=(sqrt5 -2)/(5-4)`

` = sqrt5 - 2`

Therefore,

`x - 1/x=sqrt5 +2 -(sqrt5 - 2)`

`= sqrt5 +2 - sqrt5 +2`

 ` = 2+2`

 ` = 4`