(a) Viết lại biểu thức:
\(P=\left[\dfrac{x-\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}-\dfrac{x}{\sqrt{x}\left(\sqrt{x}-2\right)}\right]:\dfrac{\sqrt{x}-1}{\sqrt{x}-2}\)
\(=\dfrac{\sqrt{x}\left(x-\sqrt{x}+2\right)-x\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}\cdot\dfrac{\sqrt{x}-2}{\sqrt{x}-1}\)
\(=\dfrac{2\sqrt{x}-2x}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{-2\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=-\dfrac{2}{\sqrt{x}+1}\) (đpcm).
(b) \(P\in Z\Rightarrow\left(\sqrt{x}+1\right)\inƯ\left(2\right)=\left\{\pm1;\pm2\right\}\).
Mà theo đề: \(x>0\Rightarrow\sqrt{x}+1>1\Rightarrow\sqrt{x}+1=2\Leftrightarrow x=1\left(L\right)\).
Vậy: Không tồn tại \(x\) để \(P\in Z\) (đpcm).
