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ĐKXĐ: ...
\(P=\left(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{2}{x}-\frac{2-x}{x\left(\sqrt{x}+1\right)}\right)\)
\(=\left(\frac{\sqrt{x}\left(\sqrt{x}+1\right)+\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\left(\frac{2\left(\sqrt{x}+1\right)-2+x}{x\left(\sqrt{x}+1\right)}\right)\)
\(=\frac{\left(x+2\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}.\frac{x\left(\sqrt{x}+1\right)}{\left(x+2\sqrt{x}\right)}=\frac{x}{\sqrt{x}-1}\)
\(x=\frac{2}{2-\sqrt{3}}=\frac{4}{4-2\sqrt{3}}=\left(\frac{2}{\sqrt{3}-1}\right)^2\)
\(\Rightarrow P=\frac{\frac{2}{2-\sqrt{3}}}{\frac{2}{\sqrt{3}-1}-1}=\frac{\frac{2}{2-\sqrt{3}}}{\frac{3-\sqrt{3}}{\sqrt{3}-1}}=\frac{2}{2\sqrt{3}-3}\)
\(\sqrt{P}\) xác định khi \(x>1\)
Khi đó: \(\sqrt{P}=\sqrt{\frac{x}{\sqrt{x}-1}}=\sqrt{\frac{x}{\sqrt{x}-1}-4+4}=\sqrt{\frac{\left(\sqrt{x}-2\right)^2}{\sqrt{x}-1}+4}\ge2\)
\(\sqrt{P}_{min}=2\) khi \(x=4\)
\(\left(x-1\right)-\frac{2x-2\sqrt{x}}{\sqrt{x}-1}+\frac{x\sqrt{x}+1}{x-\sqrt{x}+1}\)
\(=\left(x-1\right)-\frac{2\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}-1}+\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{x-\sqrt{x}+1}\)
\(=x-1-2\sqrt{x}+\sqrt{x}+1\)
\(=x-\sqrt{x}\)
ĐKXĐ:\(\sqrt{x}\ge0\Leftrightarrow x\ge0\)
Rút gọn: P=\(\left(\frac{\sqrt{x}-2}{x-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\frac{\left(1-x\right)^2}{2}=\left(\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(x-1\right)\left(\sqrt{x}+1\right)}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(x-1\right)\left(\sqrt{x}+1\right)}\right).\frac{\left(x-1\right)^2}{2}\)
\(=\frac{x+\sqrt{x}-2\sqrt{x}-2-x+\sqrt{x}-2\sqrt{x}+2}{\left(x-1\right)\left(\sqrt{x}+1\right)}.\frac{\left(x-1\right)^2}{2}=\frac{2\sqrt{x}\left(x-1\right)^2}{2\left(x-1\right)\left(\sqrt{x}+1\right)}=\sqrt{x}\left(\sqrt{x}-1\right)=x-1\)