What is the greatest possible value of $x+y$ such that $x^{2} + y^{2} =90$ and $xy=27$?
I recall that $(x+y)^{2} = x^{2} + 2xy + y^{2}$, so I can substitute the given values of $x^{2}$ and $xy$ into this identity and get $(x+y)^{2} = 90 + 2 \cdot 27 = 144$. Taking the square root of both sides, I get $x+y = \pm 12$. However, I want the greatest possible value of $x+y$, so I choose the positive sign and get $x+y = 12$.

