If `x=(sqrt(3)+sqrt(2))/(sqrt(3)-sqrt(2))` and `y=(sqrt(3)-sqrt(2))/(sqrt(3)+sqrt(2))` , find `x^2+y^2`

Explanation

`x=(sqrt3+sqrt2)/(sqrt3-sqrt2)`

After rationalizing

`x=(sqrt3+sqrt2)/(sqrt3-sqrt2)*(sqrt3+sqrt2)/(sqrt3+sqrt2)`

`x=(sqrt3+sqrt2)^2/(3-2)=3+2+2sqrt6`

`x=5+2sqrt6`

Similarly,

`y=5-2sqrt6`

`x^2+y^2=(5+2sqrt6)^2+(5-2sqrt6)^2`

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