Question

If `x = sqrt(2) + 1` then `x + 1/x` is equal to ______.

Options

• 2

• `2sqrt(2)`

• `sqrt(2) - 1`

• 4

Answer 1

If `x = sqrt(2) + 1` then `x + 1/x` is equal to `underlinebb(2sqrt(2))`.

Explanation:

Given `x = sqrt(2) + 1`, we want to find `x + 1/x`.

Step 1: Calculate `1/x`: 

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

Multiply numerator and denominator by the conjugate `sqrt(2) - 1`:

`1/(sqrt(2) + 1) xx (sqrt(2) - 1)/(sqrt(2) - 1)`

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

= `(sqrt(2) - 1)/(2 - 1)`

= `sqrt(2) - 1`

Step 2: Add x and `1/x`:

`x + 1/x`

= `(sqrt(2) + 1) + (sqrt(2) - 1)`

= `sqrt(2) + 1 + sqrt(2) - 1`

= `2sqrt(2)`

Therefore, `x + 1/x = 2sqrt(2)`.