Question

\(\dfrac{2}{\sqrt{x}-1}+\dfrac{2}{\sqrt{x}+1}-\dfrac{5-\sqrt{x}}{x-1}\)

Answer 1

\(\dfrac{2}{\sqrt{x}-1}+\dfrac{2}{\sqrt{x}+1}-\dfrac{5-\sqrt{x}}{x-1}\)

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

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

\(=\dfrac{4\sqrt{x}}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{5-\sqrt{x}}{x-1}\)

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

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

\(=\dfrac{5}{\sqrt{x}+1}\)