Q13/089) Find the real solution(s) to the equation 36/x^(1/2) +9/y^(1/4) = 42−9x^(1/2) −y^(1/2)

We have

36/x^(1/2) + 9x^(1/2) +     9/y^(1/4)+ y^(1/2) = 42

Complete the square for x and y terms giving

36/x^(1/2) + 9x^(1/2) - 36  +    9/y^(1/4)+ y^(1/2) = 42

Hence 9( 2/x^1/4 – x^(1/4))^2 + (3/y^(1/4) – y^(1/4))^2 = 0
so  2/x^(1/4) – x^(1/4) = 0 or x = 2^2 = 4
And 3/y^(1/4) – y^(1/4) = 0 or y = 9