\(y\ge1\)
\(x^2+3x-xy^2-y^2+2=0\)
\(\Leftrightarrow x^2+x-\left(y^2-2\right)x-\left(y^2-2\right)=0\)
\(\Leftrightarrow x\left(x+1\right)-\left(y^2-2\right)\left(x+1\right)=0\)
\(\Leftrightarrow\left(x+1\right)\left(x-y^2+2=0\right)\Rightarrow\left[{}\begin{matrix}x=-1\\x=y^2-2\end{matrix}\right.\)
- Với \(x=-1\Rightarrow y-1-4\sqrt{y-1}=0\)
\(\Leftrightarrow\sqrt{y-1}\left(\sqrt{y-1}-4\right)=0\Rightarrow\left[{}\begin{matrix}\sqrt{y-1}=0\\\sqrt{y-1}=4\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}y=1\\y=17\end{matrix}\right.\)
- Với \(x=y^2-2\Rightarrow y^2+y-4\sqrt{y-1}-2=0\)
Đặt \(\sqrt{y-1}=a\ge0\Rightarrow\left\{{}\begin{matrix}y=a^2+1\\y^2=a^4+2a^2+1\end{matrix}\right.\)
\(a^4+2a^2+1+a^2+1-4a-2=0\)
\(\Leftrightarrow a^4+3a^2-4a=0\)
\(\Leftrightarrow a\left(a^3+3a-4\right)=0\)
\(\Leftrightarrow a\left(a-1\right)\left(a^2+a+4\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}a=0\\a=1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\sqrt{y-1}=0\\\sqrt{y-1}=1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}y=1\Rightarrow x=-1\\y=2\Rightarrow x=2\end{matrix}\right.\)
