jmc

Using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, we can find the roots of the quadratic. We find that $x=\frac{-5\pm \sqrt{25-4c}}{2}$. Thus, setting our two expressions for $x$ equal to each other, we find that $$\begin{array}{rl}\frac{-5\pm \sqrt{25-4c}}{2}& =\frac{-5\pm \sqrt{c}}{2}\Rightarrow \\ 25-4c& =c\Rightarrow \\ c& =\boxed{5}.\end{array}$$