I have a monic quadratic equation $x^2 + bx + c$, where $b, c \in \mathbb{Z}$ for which I need to compute the roots. $x$ is also an integer. However, I don't have the exact value of $b$, only an approximation. For concreteness, say I know the most significant $\frac{3}{4}$ digits of $b$, but not the rest. Is there any way to find the roots that's faster than trying every value in the range given by the approximation?
EDIT: I also know that $b \leq \sqrt{c}$.