Find the integer integers k for which there exists an integer x $\sqrt{39-6\sqrt{12}}+\sqrt{kx(kx+\sqrt{12}+3)}=2k$
So far I haven't advanced much. Removing brackets didn't do anything for me and was pretty bashy. I am actually lost on this problem, if only the three wasn't in the bracket....Any hint will be well appreciated
Hint: Isolate $\sqrt{kx(kx+\sqrt{12}+3)}$ on one side of the equation, and square both sides. After a bit of simplification, you get an equation of the form $A(k,x) + B(k,x) \sqrt{3} = 0$, where $A(k,x)$ and $B(k,x)$ must be integers. This tells you that $A(k,x)$ and $B(k,x)$ must both be $0$.