ĐKXĐ: \(\left\{{}\begin{matrix}x>=0\\x\notin\left\{1;4\right\}\end{matrix}\right.\)
\(E=\left(\dfrac{1}{\sqrt{x}-1}-1\right):\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)
\(=\dfrac{1-\sqrt{x}+1}{\sqrt{x}-1}:\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}+2\right)\cdot\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{-\left(\sqrt{x}-2\right)}{\sqrt{x}-1}\cdot\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{x-1-x+4}\)
\(=-\dfrac{1}{3}\left(\sqrt{x}-2\right)^2\)
Để E nguyên thì \(\left(\sqrt{x}-2\right)^2⋮3\)
=>\(\left(\sqrt{x}-2\right)^2⋮9\)(Vì \(\left(\sqrt{x}-2\right)^2\) là số bằng bình phương của một số
=>\(\sqrt{x}-2=3k\)
=>\(\sqrt{x}=3k+2\)
=>\(x=\left(3k+2\right)^2\)
