Cho biểu thức Q= ( căn x/ căn x-1+ 1/ x- căn x): ( 1/ căn x+ 1+ 2/ x-1) ( với x>0, x khác 1) a)    Rút gọn biểu thức Q

Trả lời

$\begin{array}{l}a)Q=\left(\frac{\sqrt{x}}{\sqrt{x}-1}+\frac{1}{x-\sqrt{x}}\right):\left(\frac{1}{\sqrt{x}+1}+\frac{2}{x-1}\right)\left(\begin{array}{l}x>0\\ x\ne 1\end{array}\right)\\ =\frac{x+1}{\sqrt{x}.\left(\sqrt{x}-1\right)}:\frac{\sqrt{x}-1+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\frac{x+1}{\sqrt{x}.\left(\sqrt{x}-1\right)}.\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}+1}=\frac{x+1}{\sqrt{x}}\end{array}$

$\begin{array}{l}b)x=3-2\sqrt{2}\Rightarrow \sqrt{x}=\sqrt{2}-1(tmdk)\\ \Rightarrow Q=\frac{3-2\sqrt{2}+1}{\sqrt{2}-1}=2\sqrt{2}\\ c)Q>2\Rightarrow \frac{\sqrt{x}+2}{\sqrt{x}}>2\iff x+1>2\sqrt{x}=x-2\sqrt{x}+1={\left(\sqrt{x-1}\right)}^2>0\end{array}$

$(đúng với mọi x>0,x\ne 1)$