\(B=\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\sqrt{x}+1}{x+\sqrt{x}+1}-\frac{1}{\sqrt{x}-1}\)
\(=\frac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\frac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\frac{\sqrt{x}}{x+\sqrt{x}+1}\)
b/ \(A.B=m\Leftrightarrow\frac{\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)}.\frac{\sqrt{x}}{\left(x+\sqrt{x}+1\right)}=m\Leftrightarrow\frac{\sqrt{x}}{\sqrt{x}-1}=m\)
\(\Leftrightarrow m\sqrt{x}-m-\sqrt{x}=0\Leftrightarrow\sqrt{x}\left(m-1\right)=m\)
- Với \(m=1\) pt vô nghiệm
- Với \(m\ne1\Rightarrow\sqrt{x}=\frac{m}{m-1}\)
Mà \(\sqrt{x}\ge0\Leftrightarrow\frac{m}{m-1}\ge0\Rightarrow\left[{}\begin{matrix}m\le0\\m>1\end{matrix}\right.\)
