The two roots that the quadratic formula gives for $x^2-10x+c=0$ differ by 6. Find $c$.

The roots are $\frac{10\pm \sqrt{100-4c}}{2}$, so the gap between them is $\frac{2\sqrt{100-4c}}{2}=\sqrt{100-4c}$. Setting $\sqrt{100-4c}=6$ gives $100-4c=36$ and $c=16$; the roots are then 8 and 2, which do differ by 6 and multiply to 16. Dividing the gap by $2a$ instead of a leads to the wrong $c=-11$, and splitting the sum 10 as 6 and 4 gives 24.