If `x=(sqrt(2)+1)/(sqrt(2)-1), y=(sqrt(2)-1)/(sqrt(2)+1)` Find the value of `x^2+y^2+x y.`

Explanation

We have
`x = (sqrt(2) +1)/(sqrt(2) -1) = ((sqrt(2) +1))/((sqrt(2) -1)) xx ((sqrt(2) -1))/((sqrt(2) -1)) = ((sqrt(2))^(2) + (1)^(2) + 2xxsqrt(2) xx1)/((sqrt(2))^(2) -(1)^(2))`
`= (2+1+2sqrt(2))/((2-1))= 3 + 2sqrt(2)`
and `y = (sqrt(2) -1)/(sqrt(2) + 1) = (sqrt(2) -1)/(sqrt(2)-1) xx ((sqrt(2) -1))/((sqrt(2)-1))= ((sqrt(2))^(2) +(1)^(2) - 2xx sqrt(2) xx1)/((sqrt(2))^(2) -(1)^(2))`
` =(2+1-2sqrt(2))/((2-1)) = 3 -2sqrt(2)`
`therefore x^(2) + y^(2) + xy =(x + y)^(2) - xy`
` = [(3 + sqrt(2) +(3 -sqrt(2))]^(2) - (3 + 2 sqrt(2))(3-2sqrt(2))`
`=(6)^(2) - [(3)^(2) - (2sqrt(2))^(2)] = 36 -(9-8)`
` = 36-1 = 35`