jmc

By the quadratic formula, the roots of the equation are $$x=\frac{-(99)\pm \sqrt{(99{)}^2-4(\frac{1}{2})c}}{2(\frac{1}{2})},$$which simplifies to $$x=-99\pm \sqrt{9801-2c}.$$This looks like our target, except that we have to get the $9801-2c$ under the radical to equal $8001$. So, we solve the equation $9801-2c=8001$, which yields $c=\boxed{900}$.