a) \(\left(\frac{1}{\sqrt{x}-1}-\frac{2}{x\sqrt{x}-x+\sqrt{x}-1}\right):\left(1-\frac{\sqrt{x}}{x+1}\right)\)
\(\Leftrightarrow\left(\frac{1}{\sqrt{x}-1}-\frac{2}{\sqrt{x}\left(\sqrt{x}-\sqrt{x}+1\right)-1}\right):\left(\frac{x+1}{x+1}-\frac{\sqrt{x}}{x+1}\right)\)
\(\Leftrightarrow\left(\frac{1}{\sqrt{x}-1}-\frac{2}{\sqrt{x}-1}\right):\left(\frac{x+1-\sqrt{x}}{x+1}\right)\)
\(\Leftrightarrow\left(\frac{-1}{\sqrt{x}-1}\right):\left(\frac{1\left(\sqrt{x}-\sqrt{x}\right)}{x+1}\right)\)
\(\Leftrightarrow\frac{-1}{\sqrt{x}-1}:\frac{1}{x+1}\)
\(\Leftrightarrow\frac{-1\left(x+1\right)}{\left(\sqrt{x}-1\right)1}\)
\(\Leftrightarrow\frac{-x-1}{\sqrt{x}-1}\)
c) \(P=\frac{-x-1}{\sqrt{x}-1}\)
Tìm x nguyên để P nguyên
Ta có:\(\frac{-\left(\sqrt{x}^2-1^2\right)}{\sqrt{x}-1}\)
\(=\frac{-\left[\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\right]}{\sqrt{x}-1}\)
\(=-\left(\sqrt{x}+1\right)\)
\(=-\sqrt{x}-1\)
P nguyên thì \(\sqrt{x}\) phải là số nguyên
=> x vô số nghiệm
