ĐKXĐ: ...
Đặt \(\left\{{}\begin{matrix}\sqrt{x+3}=u\ge0\\\sqrt{y+3}=v\ge0\end{matrix}\right.\) ta được:
\(\left\{{}\begin{matrix}u^2-2=\sqrt{v+2}\\v^2-2=\sqrt{u+2}\end{matrix}\right.\)
\(\Rightarrow u^2-v^2=\sqrt{v+2}-\sqrt{u+2}\)
\(\Leftrightarrow\left(u-v\right)\left(u+v\right)+\dfrac{u-v}{\sqrt{u+2}+\sqrt{v+2}}=0\)
\(\Leftrightarrow\left(u-v\right)\left(u+v+\dfrac{1}{\sqrt{u+2}+\sqrt{v+2}}\right)=0\)
\(\Leftrightarrow u-v=0\Leftrightarrow u=v\)
Thế vào pt đầu:
\(u^2-2=\sqrt{u+2}\)
Đặt \(\sqrt{u+2}=t>0\Rightarrow2=t^2-u\)
\(\Rightarrow u^2-\left(t^2-u\right)=t\)
\(\Rightarrow u^2-t^2+u-t=0\)
\(\Leftrightarrow\left(u-t\right)\left(u+t+1\right)=0\)
\(\Leftrightarrow u=t\Leftrightarrow u=\sqrt{u+2}\)
\(\Leftrightarrow u^2-u-2=0\Leftrightarrow u=2\)
\(\Leftrightarrow\sqrt{x+3}=2\Rightarrow x=y=1\)
