\(P=\left(\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right).\left(\frac{x\left(\sqrt{x}+1\right)}{\sqrt{x}-3}\right)\)
\(=\frac{x-2\sqrt{x}+1-\left(x-\sqrt{x}-2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(\sqrt{x}\left(\sqrt{x}+1\right)\right)}{\sqrt{x}-3}\)
\(=\frac{-\left(\sqrt{x}-3\right).\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}=\frac{\sqrt{x}}{1-\sqrt{x}}\)
Để \(P< -1\Rightarrow\frac{\sqrt{x}}{1-\sqrt{x}}< -1\)
\(\Rightarrow\frac{\sqrt{x}}{1-\sqrt{x}}+1< 0\Rightarrow\frac{\sqrt{x}+1-\sqrt{x}}{1-\sqrt{x}}< 0\)
\(\Rightarrow\frac{1}{1-\sqrt{x}}< 0\Rightarrow1-\sqrt{x}< 0\Rightarrow x>1\)
Kết hợp ĐKXĐ \(\Rightarrow\left\{{}\begin{matrix}x>1\\x\ne9\end{matrix}\right.\)
\(P=\frac{\sqrt{x}-1+1}{1-\sqrt{x}}=-1+\frac{1}{1-\sqrt{x}}\)
Để P nguyên \(\Rightarrow\frac{1}{1-\sqrt{x}}\) nguyên \(\Rightarrow1-\sqrt{x}=Ư\left(1\right)=\left\{-1;1\right\}\)
\(\Rightarrow\left[{}\begin{matrix}1-\sqrt{x}=-1\\1-\sqrt{x}=1\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\sqrt{x}=2\\\sqrt{x}=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=4\\x=0\left(l\right)\end{matrix}\right.\)
