Integer Solutions for $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$

[anemone]

Find all integer solutions to the equation $4x+y+4\sqrt{xy}-28\sqrt{x}-14\sqrt{y}+48=0$.

 

[anemone]

Spoiler: Solution of other

From the given equation we get

$\begin{align*}(2\sqrt{x})^2+2(2\sqrt{x})\sqrt{y}+(\sqrt{y})^2-28\sqrt{x}-14\sqrt{y}+48&=0\\ (2\sqrt{x}+\sqrt{y})^2-14(2\sqrt{x}+\sqrt{y})+48&=0\\ (2\sqrt{x}+\sqrt{y}-6)(2\sqrt{x}+\sqrt{y}-8)&=0\end{align*}$

Thus, $2\sqrt{x}+\sqrt{y}=6$ or $2\sqrt{x}+\sqrt{y}=8$. The first of these equations is satisfied by the integer pairs $(x,\,y)\in {(0,\,36),\,(1,\,16),\,(4,\,4),\,(9,\,0)}$ and the second is satisfied by $(x,\,y)\in {(0,\,64),\,(1,\,36),\,(4,\,16),\,(9,\,4),\,(16,\,0)}$.

Thus the complete solution set is

${(0,\,36),\,(1,\,16),\,(4,\,4),\,(9,\,0),\,(0,\,64),\,(1,\,36),\,(4,\,16),\,(9,\,4),\,(16,\,0)}$