Đặt \(\left\{{}\begin{matrix}\sqrt{x+y}=a\ge0\\\sqrt{x-y}=b\ge0\end{matrix}\right.\)
\(\left\{{}\begin{matrix}a-b=2\\\sqrt{\frac{a^4+b^4}{2}}+ab=4\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a-b=2\\\sqrt{\frac{a^4+b^4}{2}}=4-ab\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a-b=2\\\frac{a^4+b^4}{2}=a^2b^2-8ab+16\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=2\\a^4+b^4-2a^2b^2=16\left(2-ab\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=2\\\left(a^2-b^2\right)^2=16\left(2-ab\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=2\\4\left(a+b\right)^2=16\left(2-ab\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=2\\\left(a+b\right)^2=8-4ab\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}b=a-2\\\left(2a-2\right)^2=8-4a\left(a-2\right)\end{matrix}\right.\)
\(\Rightarrow4a^2-8a+4=8-4a^2+8a\)
\(\Rightarrow2a^2-4a-1=0\Rightarrow\left[{}\begin{matrix}a=\frac{2+\sqrt{6}}{2}\\a=\frac{2-\sqrt{6}}{2}\left(l\right)\end{matrix}\right.\) \(\Rightarrow b=\frac{\sqrt{6}-2}{2}\)
\(\Rightarrow\left\{{}\begin{matrix}\sqrt{x+y}=\frac{2+\sqrt{6}}{2}\\\sqrt{x-y}=\frac{\sqrt{6}-2}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x+y=\frac{5+2\sqrt{6}}{2}\\x-y=\frac{5-2\sqrt{6}}{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=\frac{5}{2}\\y=\sqrt{6}\end{matrix}\right.\)