A fraction problem $$a = x + \frac1x \\b =y + \frac1y \\ c = xy +\frac1{xy} $$ 
Express $c$ in terms of $a$ and $b$

Note that $c$ is not uniquely specified by $a$ and $b$, since solving $a=x+\frac{1}{x}$ for $x$ yields two solutions which are reciprocals of each other, and applying $x\mapsto\frac{1}{x}$ or $y\mapsto\frac{1}{y}$ to $xy+\frac{1}{xy}$ yields $\frac{x}{y}+\frac{y}{x}$. However, applying either substitution again gives back $xy+\frac{1}{xy}$, so $c$ can take on either of those values.
Let $c_1=xy+\frac{1}{xy}$ and $c_2=\frac{x}{y}+\frac{y}{x}$. One can check that
$$c_1+c_2=ab$$
and
$$c_1c_2=a^2+b^2-4.$$
Use the quadratic formula to solve for the possible values of $c$.
