TITLE: Finding two numbers given their sum and their product
QUESTION [11 upvotes]: Which two numbers when added together yield $16$, and when multiplied together yield $55$.

I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't. 
In $x+y=16$, I know $x=16/y$ but when I plug it back in I get something like $16/y + y = 16$, then I multiply the left side by $16$ to get $2y=256$ and then ultimately $y=128$. Am I doing something wrong?

REPLY [3 votes]: If $x+y=16$ and $xy=55$, then, since $\dfrac{x+y}{2}=8$, we can let $x = 8-z$ and $y=8+z$ and it won't hurt to suppose that $z$ is positive..
Then
\begin{align}
   xy &= 55 \\
   (8-z)(8+z) &= 55 \\
   64 - z^2 &= 55 \\
   z^2 &= 9 \\
   z &= 3 \\
   x &= 8-3 = 5 \\
   y &= 8+3 = 11
\end{align}