How to check, when/whether a quadratic equation has iteger roots I am familliar with the Jacobi symbol and thus can check for an equation, whether it has solutions in a given $\mathbb{Z}_n$. But how do I check for integer solutions in general?
In perticular, I found the question: for which $a$ does $ax^2+2x+4=0$ have integer solutions?

We interpret the question as asking for which integers $a$ does the equation have at least one integer solution. Of course there is $a=0$. Now look for non-zero $a$.
By the Rational Roots Theorem, any rational solution is of the form $\frac{p}{q}$, where $p$ divides $4$ and $q$ divides $a$. If this is to be an integer, it must be $\pm 1$, $\pm 2$, or $\pm 4$. Check for each of these whether it yields an integer value of $a$.
