Finding condition for integral roots of a quadratic equation. I need to find the values of k(possible) for which the quadratic equation $$x^2+2kx+k =0$$ will have integral roots.
So I assumed roots to be $a,b$
Then I got the condition $a+b=-2k$and $a\cdot b=k$; so combining these I get $a+b+2ab=0$;
And now I need to find the integral values of $a,b$ for which this equation is satisfied,how should I procced from here??
Also is there any shorter much elegant way to do this question.
(Note-A hint would suffice)

Here is a hint to develop your existing method. Multiply the equation in $a$ and $b$ by $2$ and add a constant which enables you to factorise it.
