a: \(P=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2}\)
\(=\left(\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right)\cdot\dfrac{2}{\sqrt{x}-1}\)
\(=\dfrac{x+2+\sqrt{x}\left(\sqrt{x}-1\right)-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)\cdot\left(\sqrt{x}-1\right)}\cdot2\)
\(=\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)^2}\cdot\dfrac{2}{x+\sqrt{x}+1}\)
\(=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)^2}\cdot\dfrac{2}{x+\sqrt{x}+1}=\dfrac{2}{x+\sqrt{x}+1}\)
b: P=2/7
=>\(\dfrac{2}{x+\sqrt{x}+1}=\dfrac{2}{7}\)
=>\(x+\sqrt{x}+1=7\)
=>\(x+\sqrt{x}-6=0\)
=>\(\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)=0\)
=>\(\sqrt{x}-2=0\)
=>x=4(nhận)
c: \(P^2-2P=P\left(P-2\right)\)
\(=\dfrac{2}{x+\sqrt{x}+1}\left(\dfrac{2}{x+\sqrt{x}+1}-2\right)\)
\(=\dfrac{2}{x+\sqrt{x}+1}\cdot\dfrac{2-2x-2\sqrt{x}-2}{x+\sqrt{x}+1}\)
\(=\dfrac{-4\left(x+\sqrt{x}\right)}{\left(x+\sqrt{x}+1\right)^2}< 0\)
=>P^2<2P
