Question

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

Answer 1

x = `sqrt(2) + sqrt(3)`

x² = `(sqrt(2) + sqrt(3))^2`

= `(sqrt(2))^2 + (sqrt(3))^2 + 2sqrt(2)*sqrt(3)`

= `2 + 3 + 2sqrt(6)`

= `5 + 2sqrt(6)`

`(x^2 + 1)/(x^2 - 2) = (5 + 2sqrt(6) + 1)/(5 + 2sqrt(6) - 2)`

= `(6 + 2sqrt(6))/(3 + 2sqrt(6))`

= `((6 + 2sqrt(6))(3 - 2sqrt(6)))/((3 + 2sqrt(6))(3 - 2sqrt(6))`

= `(18 - 12sqrt(6) + 6sqrt(6) - (2sqrt(6))^2)/(3^2 - (2sqrt(6))^2`

= `(18 - 6sqrt(6) - 4 xx 6)/(9 - 4 xx 6)`

= `(18 - 24 - 6sqrt(6))/(9 - 24)`

= `(-6 - 6sqrt(6))/(-15)`

= `(- 3(2 + 2sqrt(6)))/(-15)`

`(x^2 + 1)/(x^2 - 2) = (2 + 2sqrt(6))/5`