Question

If x =`[sqrt5 - 2 ]/[ sqrt5 + 2]` and y = `[ sqrt5 + 2]/[ sqrt5 - 2]`; find:

x² + y² + xy.

Answer 1

We rationalize the denominator,

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

`x = (5 + 4 - 3sqrt5)/ (5 -4)`

`x = (9 - 4 sqrt5)/1`

Then, `x^2 = (9 - 4 sqrt5) (9 - 4 sqrt5)`

`= 81 + 16 ×5 - 72sqrt5 `

`= 161 - 72sqrt5`

We rationalize the denominator,

`y = (sqrt5 + 2)/(sqrt5 - 2) xx (sqrt5 + 2)/(sqrt5 + 2)`

`y = (5 + 4 + 4sqrt5)/(5-4)`

`y = (9 + 4sqrt5)/1`

Then, `y^2 = (9 + 4sqrt5) (9 + 4 sqrt5)`

`= 81 + 16 + 5 + 72 sqrt5`

`= 161 + 72sqrt5`

Now, `x = (sqrt5 + 2)/(sqrt5 - 2)`

and `y = (sqrt5 - 2)/(sqrt5 + 2)`

Then, `xy = (sqrt5 + 2)/(sqrt5 - 2) xx (sqrt5 - 2)/(sqrt5 + 2)`

xy = 1

Therefore, `x^2 + y^2 + xy`

`=161 - 72sqrt5 + 161 + 72sqrt5 + 1 = 323`