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\(M=\frac{x+\sqrt{x^2-2x}}{x-\sqrt{x^2-2x}}-\frac{x-\sqrt{x^2-2x}}{x+\sqrt{x^2-2x}}\left(x< 0;x\ge2\right)\)
\(=\frac{\left(x+\sqrt{x^2-2x}\right)\left(x+\sqrt{x^2-2x}\right)}{x^2-\sqrt{x^2-2x}^2}-\frac{\left(x-\sqrt{x^2-2x}\right)\left(x-\sqrt{x^2-2x}\right)}{x^2-\sqrt{x^2-2x}^2}\)
\(=\frac{x^2+x\sqrt{x^2-2x}+x\sqrt{x^2-2x}+x^2-2x}{x^2-x^2-2x}-\frac{x^2-x\sqrt{x^2-2x}-x\sqrt{x^2-2x}+x^2-2x}{x^2-x^2-2x}\)
\(=\frac{2x^2+2x\sqrt{x^2-2x}-2x}{-2x}-\frac{2x^2-2\sqrt{x^2-2x}-2x}{-2x}\)
\(=\frac{2x^2+2x\sqrt{x^2-2x}-2x-2x^2+2x\sqrt{x^2-2x}+2x}{-2x}\)
\(=\frac{4x\sqrt{x^2-2x}}{-2x}=-2x\sqrt{x^2-2x}\)
\(A=\frac{x+\sqrt{x^2-2x}}{x-\sqrt{x^2-2x}}-\frac{x-\sqrt{x^2-2x}}{x+\sqrt{x^2+2x}}\)
\(=\frac{\left(x+\sqrt{x^2-2x}\right)^2-\left(x-\sqrt{x^2-2x}\right)^2}{x^2-\left(\sqrt{x^2-2x}\right)^2}\)
\(=\frac{x^2+x^2-2x+2x\sqrt{x^2-2x}-\left(x^2+x^2-2x-2x\sqrt{x^2-2x}\right)}{x^2-\left(x^2-2x\right)}\)
\(=\frac{2x^2-2x-2x^2+2x+2x\sqrt{x^2-2x}+2x\sqrt{x^2-2x}}{2x}\)
\(=\frac{4x\sqrt{x^2-2x}}{2x}=2\sqrt{x^2-2x}\)
\(\frac{A}{\sqrt{2}}\)=\(\frac{\sqrt{x-1+2\sqrt{x-1}+1}+\sqrt{x-1-2\sqrt{x-1}+1}}{\sqrt{2x-1+2\sqrt{2x-1}+1}-\sqrt{2x-1-2\sqrt{2x-1}+1}}\) (DK \(x\ge1\)
\(=\frac{\sqrt{x-1}+1+\left|\sqrt{x-1}-1\right|}{\sqrt{2x-1}+1-\left|\sqrt{2x-1}-1\right|}\)
vs \(x\ge2\) \(\frac{\sqrt{x-1}+1+\sqrt{x-1}-1}{\sqrt{2x-1}+1-\sqrt{2x-1}+1}=\frac{2\sqrt{x-1}}{2}=\sqrt{x-1}\) \(\Rightarrow A=\sqrt{2x-2}\)
vs \(1\le x< 2\) \(\frac{\sqrt{x-1}+1+1-\sqrt{x-1}}{\sqrt{2x-1}+1-1+\sqrt{2x-1}}=\frac{1}{\sqrt{2x-1}}\) \(\Rightarrow A=\frac{\sqrt{2}}{\sqrt{2x-1}}\)
\(\sqrt{2X-1}\ge1\Leftrightarrow X\ge1\)NEN SUY RA THEO CACH LAM CUA TO
THOI U AM BUSY SEE YOU AGAIN
\(A=\left(a-1\right)\sqrt{\frac{a}{a-1}}+\sqrt{a\left(a-1\right)}-a\sqrt{\frac{a-1}{a}}\)
\(A=\sqrt{\left(a-1\right)^2.\frac{a}{a-1}}+\sqrt{a\left(a-1\right)}-\sqrt{a^2.\frac{a-1}{a}}\)
\(A=\sqrt{\left(a-1\right)a}+\sqrt{a\left(a-1\right)}-\sqrt{a\left(a-1\right)}\)
\(A=\sqrt{a\left(a-1\right)}\)
\(A=\frac{\left(x+\sqrt{x^2-2x}\right)^2-\left(x-\sqrt{x^2-2x}\right)^2}{\left(x-\sqrt{x^2-2x}\right)\left(x+\sqrt{x^2-2x}\right)}\)
\(=\frac{2x\times2\sqrt{x^2-2x}}{2x}=2\sqrt{x^2-2x}\)