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\(A=\dfrac{2}{x-1}\sqrt{\dfrac{\left(x-1\right)^2}{4x^2}}=\dfrac{2}{x-1}\left|\dfrac{x-1}{2x}\right|=\dfrac{\left|x-1\right|}{\left(x-1\right)\left|x\right|}\)
\(B=\left(x^2-4\right)\sqrt{\dfrac{9}{x^2-4x+4}}=\dfrac{3\left(x^2-4\right)}{\left|x-2\right|}\)
a) Ta có: \(A=\dfrac{2}{x-1}\cdot\sqrt{\dfrac{x^2-2x+1}{4x^2}}\)
\(=\dfrac{2}{x-1}\cdot\dfrac{x-1}{2x}\)
\(=\dfrac{1}{x}\)
b) Ta có: \(\left(x^2-4\right)\cdot\sqrt{\dfrac{9}{x^2-4x+4}}\)
\(=\dfrac{\left(x-2\right)\left(x+2\right)\cdot3}{\left(x-2\right)^2}\)
\(=\dfrac{3x+6}{x-2}\)
`[2x+\sqrt{2}]/[4x^2+4\sqrt{2}x+\sqrt{2}]`
`=[\sqrt{2}(\sqrt{2}x+1)]/[\sqrt{2}(2\sqrt{2}x^2+4x+1)]`
`=[\sqrt{2}x+1]/[2\sqrt{2}x^2+4x+1]`
ĐK: \(x\ge\frac{1}{4}\)
Ta có \(A^2=4x+2\sqrt{4x^2-\left(4x-1\right)}=4x+2\sqrt{\left(2x-1\right)^2}\)
Với \(x\ge\frac{1}{2},A=4x+2\left(2x-1\right)=8x-2\)
Do \(A\ge0\) nên \(A=\sqrt{8x-2}\)
Với \(\frac{1}{4}\le x< \frac{1}{2},A^2=4x+2\left(1-2x\right)=2\)
Do \(A\ge0\) nên \(A=\sqrt{2}\)
\(=\sqrt{4x-1-2\sqrt{4x-1}+1}+\sqrt{4x-1+2\sqrt{4x-1}+1}\)
\(=\sqrt{\left(\sqrt{4x-1}-1\right)^2}+\sqrt{\left(\sqrt{4x-1}+1\right)^2}\)
\(=\left|\sqrt{4x-1}-1\right|+\sqrt{4x-1}+1\)
\(=\left[{}\begin{matrix}2\sqrt{4x-1}\text{ nếu }x\ge\dfrac{1}{2}\\2\text{ nếu }\dfrac{1}{4}\le x< \dfrac{1}{2}\end{matrix}\right.\)
A=2x-|2x+1|
TH1: x>=-1/2
A=2x-2x-1=-1
TH2: x<-1/2
A=2x+2x+1=4x+1