Almost Vieta's

Algebra Level 5

$\begin{cases} a+b+c =10\\ ab+bc+ca=25 \end{cases}$

Let $a$ , $b$ and $c$ be real numbers such that the system of equations above is true. If the sum of the minimum and maximum values of $c$ can be written in the form $\frac{x}{y}$ , where $x$ and $y$ are coprime positive integers, find $x+y$ .