ĐKXĐ: \(x>0;x\ne1\)
\(P=\left(\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\left(\dfrac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\dfrac{\sqrt{x}-1+2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+1}{\sqrt{x}}.\left(\sqrt{x}-1\right)\)
\(=\dfrac{x-1}{\sqrt{x}}\)
Để \(P< 0\Leftrightarrow\dfrac{x-1}{\sqrt{x}}< 0\)
\(\Rightarrow x-1< 0\Leftrightarrow x< 1\)
Kết hợp ĐKXĐ: \(\Rightarrow0< x< 1\)
Khi \(x=4-2\sqrt{3}=\left(\sqrt{3}-1\right)^2\Rightarrow P=\dfrac{4-2\sqrt{3}-1}{\sqrt{\left(\sqrt{3}-1\right)^2}}=\dfrac{3-2\sqrt{3}}{\sqrt{3}-1}=\dfrac{\sqrt{3}-3}{2}\)
