điều kiện xác định : \(x>0;x\ne1\)
ta có : \(A=\dfrac{\left(\sqrt{x^2+4}-2\right)\left(x+\sqrt{x}+1\right)\left(\sqrt{x^2+4}+2\right)\sqrt{x-2\sqrt{x}+1}}{x\left(x\sqrt{x}-1\right)}\)
\(\Leftrightarrow A=\dfrac{\left(x^2+4-4\right)\left(x+\sqrt{x}+1\right)\sqrt{\left(\sqrt{x}-1\right)^2}}{x\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)\(\Leftrightarrow\left[{}\begin{matrix}A=\dfrac{-x^2}{x}=-x\left(x< 1\right)\\A=\dfrac{x^2}{x}=x\left(x>1\right)\end{matrix}\right.\)
để \(A\ge0\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}-x\ge0\\x< 1\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge0\\x>1\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x\le0\left(L\right)\\x>1\left(N\right)\end{matrix}\right.\)
vậy \(x>1\)
ĐK: \(\left\{{}\begin{matrix}x\ge0\\x\ne1\end{matrix}\right.\)
\(A=\dfrac{\left(\sqrt{x^2+4}-2\right)\left(x+\sqrt{x}+1\right)\left(\sqrt{x^2+4}+2\right)\left(x-2\sqrt{x}+1\right)}{x\left(x\sqrt{x}-1\right)}\)
\(A=\dfrac{\left(x+\sqrt{x}+1\right)\left(x^2+4-4\right)\left(\sqrt{x}-1\right)^2}{x\left[\left(\sqrt{x}\right)^3-1\right]}\)
\(A=\dfrac{\left(x+\sqrt{x}+1\right)x^2\left(\sqrt{x}-1\right)^2}{x\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(A=x\left(\sqrt{x}-1\right)\)
Để A \(\ge0\)
thì TH1: \(\left\{{}\begin{matrix}x\le0\\\sqrt{x}\le1\end{matrix}\right.\) \(\)\(\Leftrightarrow\left\{{}\begin{matrix}x\le0\\\sqrt{x}-1\le0\end{matrix}\right.\) \(\Rightarrow x\le0\)
Do \(x\ge0\) nên TH1 loại
TH2: \(\left\{{}\begin{matrix}x\ge0\\\sqrt{x}\ge1\end{matrix}\right.\) \(\Rightarrow x\ge1\)
Do x khác 1 nên x>1
Vậy để A\(\ge0\) thì x>1
