[tex]\sqrt{y}+\sqrt{z}=\sqrt{x+2\sqrt{3}}\Rightarrow y+z+2\sqrt{yz}=x+2\sqrt{3}\Rightarrow 2(\sqrt{yz}-\sqrt{3})=x-y-z\in \mathbb{Z}\Rightarrow \sqrt{yz}-\sqrt{3}\in \mathbb{Q}[/tex]
Đặt [tex]\sqrt{yz}-\sqrt{3}=k\Rightarrow \sqrt{yz}=k+\sqrt{3}\Rightarrow yz=k^2+2\sqrt{3}k+3\Rightarrow 2\sqrt{3}k=yz-k^2-3\in \mathbb{Q}\Rightarrow k=0\Rightarrow yz=3\Rightarrow x=4;y=1;z=3 hoặc x=4;y=3;z=1[/tex]