For how many integer values of \(a\) does the equation \(x^2 + ax + 8a=0\) have integer solutions for \(x\)?
Note that the roots are \(x = {-a \pm \sqrt{a^2-32a} \over 2}\). So the discriminant must be a perfect square. We can factor as a(a-32). So both a and a-32 are perfect squares. You can do casework from here to get a=-49,-18,-4,0,32,36,50,81. We need not to check these, as if a is even, then sqrt(a^2-32a) is even and vice versa. Our answer is then 8.
