For how many integer values of a does the equation x^2 + ax + 6a = 0 have integer solutions for x?
+421 Firstly, we factor this equation as $a(x + 6) = -x^2.$ Because $-x^2 \ne (-x)^2,$ we have $a(x+6)=-1(x^2),$ so $a(-x-6)=x^2.$
There are 18 integer solutions. Only 1 is positive for a, and two are non negative.
