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 \(x+y\).