ĐKXĐ: ...
\(A=\left(\frac{2x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\frac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\left(\frac{x+\sqrt{x}+1-x+2}{x+\sqrt{x}+1}\right)\)
\(=\left(\frac{2x+1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right):\left(\frac{\sqrt{x}+3}{x+\sqrt{x}+1}\right)\)
\(=\frac{\left(x-\sqrt{x}\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}.\frac{\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\frac{\sqrt{x}}{\sqrt{x}+3}\)
b/ \(x=\frac{2-\sqrt{3}}{2}=\frac{4-2\sqrt{3}}{4}=\left(\frac{\sqrt{3}-1}{2}\right)^2\Rightarrow\sqrt{x}=\frac{\sqrt{3}-1}{2}\)
\(\Rightarrow A=\frac{\frac{\sqrt{3}-1}{2}}{\frac{\sqrt{3}-1}{2}+3}=\frac{\sqrt{3}-1}{\sqrt{3}+5}\)
c/ \(A=\frac{\sqrt{x}}{\sqrt{x}+3}=1-\frac{3}{\sqrt{x}+3}\)
\(A\in Z\Leftrightarrow\frac{3}{\sqrt{x}+3}\in Z\Rightarrow\sqrt{x}+3=Ư\left(3\right)\)
Mà \(\sqrt{x}+3\ge3\Rightarrow\sqrt{x}+3=3\Rightarrow x=0\)
d/ \(A=1-\frac{3}{\sqrt{x}+3}\)
Mà \(\sqrt{x}+3\le3\Rightarrow1-\frac{3}{\sqrt{x}+3}\ge1-\frac{3}{3}=0\)
\(\Rightarrow A_{min}=0\) khi \(x=0\)
