Question

Answer 1

\(A+B=\dfrac{2}{\sqrt{x}-2}+\dfrac{\sqrt{x}}{x+1}-\dfrac{4\sqrt{x}+2}{x\sqrt{x}-2x+\sqrt{x}-2}\)

\(\Rightarrow A+B=\dfrac{2\left(x+1\right)}{\left(\sqrt{x}-2\right)\left(x+1\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(x+1\right)}-\dfrac{4\sqrt{x}+2}{x\left(\sqrt{x}-2\right)+\sqrt{x}-2}\)

\(\Rightarrow A+B=\dfrac{2x+2}{\left(\sqrt{x}-2\right)\left(x+1\right)}+\dfrac{x-2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(x+1\right)}-\dfrac{4\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(x+1\right)}\)

\(\Rightarrow A+B=\dfrac{2x+2+x-2\sqrt{x}-4\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(x+1\right)}\)

\(\Rightarrow A+B=\dfrac{3x-6\sqrt{x}}{\left(\sqrt{x}-2\right)\left(x+1\right)}\)

\(\Rightarrow A+B=\dfrac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(x+1\right)}\)

\(\Rightarrow A+B=\dfrac{3\sqrt{x}}{x+1}\)

 

Answer 2

\(P=A+B=\dfrac{2x+2+x-2\sqrt{x}-4\sqrt{x}-2}{\left(x+1\right)\left(\sqrt{x}-2\right)}\)

\(=\dfrac{3x-6\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-2\right)}=\dfrac{3\sqrt{x}}{x+1}\)