jmc

In order for this quadratic to have at least one real solution, its discriminant must be non-negative. In other words, $b^2-4ac=5^2-4(2)(c)=25-8c\ge 0$. Rearranging, we have $25\ge 8c$. Dividing by 8, we have $25/8\ge c$. Therefore, the largest possible value of $c$ such that this quadratic has a real solution is $\boxed{\frac{25}{8}}$.