Question

\(\left(\dfrac{2+\sqrt{X}}{X+\sqrt{X}+1}-\dfrac{\sqrt{X}-2}{x-1}\right)×\dfrac{X\sqrt{X}+x-\sqrt{X}-1}{\sqrt{X}}\left(X>0\:,x\ne1\right)\)

Answer 1

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

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

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

\(=\dfrac{3x}{\left(x-1\right)\left(x+\sqrt{x}+1\right)}\times\dfrac{\left(\sqrt{x}+1\right)\left(x-1\right)}{\sqrt{x}}\)

\(=\dfrac{3\sqrt{x}\left(\sqrt{x}+1\right)}{x+\sqrt{x}+1}\)