Question

b) Rút gọn biểu thức \( P = \left( \frac{1}{\sqrt{x-2}} + \frac{1}{\sqrt{x+2}} + \frac{x}{\sqrt{x-4}} \right) \cdot \frac{\sqrt{x+1}}{\sqrt{x-2}} \) với \( x \geq 0 \) và \( x \neq 4 \).

Answer 1

Ta có: \(P=\left(\dfrac{1}{\sqrt{x}-2}+\dfrac{1}{\sqrt{x}+2}+\dfrac{x}{x-4}\right):\dfrac{\sqrt{x}+1}{\sqrt{x}-2}\)

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

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

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