Question

If x = `((sqrt(3) + 1))/((sqrt(3) - 1)` and y = `((sqrt(3) - 1))/((sqrt(3) + 1)`, find the values of 

x² + y²

Answer 1

x² + y²

x² + y² = (x + y)² - 2xy  ----(1)

∴ (x + y) = `((sqrt(3) + 1))/((sqrt(3) - 1)) + ((sqrt(3) - 1))/((sqrt(3) - 1)`

= `((sqrt(3) + 1)^2 + (sqrt(3) - 1)^2)/(3 - 1)`

= `(3 + 1 + 2sqrt(3) + 3 + 1 - 2sqrt(3))/(2)`

= `(8)/(2)`

= 4

and xy = `((sqrt(3) + 1))/((sqrt(3) - 1)) xx ((sqrt(3) - 1))/((sqrt(3) + 1)`

= 1

substituting in (1), we get
x² + y²
= (x + y)² - 2xy
= 16 - 2
= 14