Question

Rút gọn biểu thức P=(x+2/x√x -1 + √x/x+√x+1 - 1/√x-1) : √x-1/2

Answer 1

\(\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right):\dfrac{\sqrt{x}-1}{2}\)

\(\Leftrightarrow\left[\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{x\sqrt{x}-1}-\dfrac{x+\sqrt{x}+1}{x\sqrt{x}-1}\right].\dfrac{2}{\sqrt{x}-1}\)

\(\Leftrightarrow\left[\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{x\sqrt{x}-1}\right].\dfrac{2}{\sqrt{x}-1}\)

\(\Leftrightarrow\dfrac{x+1-2\sqrt{x}}{x\sqrt{x}-1}.\dfrac{2}{\sqrt{x}-1}\)

\(\Leftrightarrow\dfrac{\left(\sqrt{x}-1\right)^2}{x\sqrt{x}-1}.\dfrac{2}{\sqrt{x}-1}\)

\(\Leftrightarrow\dfrac{\left(\sqrt{x}-1\right)2}{x+\sqrt{x}+1}\)

\(\Leftrightarrow\dfrac{2\sqrt{x}-2}{x+\sqrt{x}+1}\)