Question

1-( \(\dfrac{x+\sqrt{x}}{\sqrt{x}+1}\) ).(\(\dfrac{1}{1-\sqrt{x}}+\dfrac{2}{\sqrt{x}+3}\) )

Answer 1

\(1-\left(\dfrac{x+\sqrt{x}}{\sqrt{x}+1}\right)\cdot\left(\dfrac{1}{1-\sqrt{x}}+\dfrac{2}{\sqrt{x}+3}\right)=1-\left(\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\right)\cdot\left(\dfrac{\sqrt{x}+3+2-2\sqrt{x}}{\left(1-\sqrt{x}\right)\left(\sqrt{x}+3\right)}\right)=1-\sqrt{x}\cdot\dfrac{5-\sqrt{x}}{\left(1-\sqrt{x}\right)\left(\sqrt{x}+3\right)}=1+\dfrac{x-5\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\dfrac{x+3\sqrt{x}-\sqrt{x}-3+x-5\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}=\dfrac{2x-\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\)