Question

Cho biểu thức P=\(\left(\dfrac{x}{x-2\sqrt{x}}+\dfrac{x}{\sqrt{x}-2}\right):\dfrac{\sqrt{x}+1}{x-4\sqrt{x}+4}\)

a Rút gọn biểu thức P

b Tìm tất cả các giá trị của x để P>0

Answer 1

a) Ta có

\(P=\left(\dfrac{x}{x-2\sqrt{x}}+\dfrac{x}{\sqrt{x}-2}\right):\dfrac{\sqrt{x}+1}{x-4\sqrt{x}+4}\) (ĐKXĐ : \(x\ne\pm4;x\ne0\) )

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

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

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

b) \(P>0\)

\(\Rightarrow\sqrt{x}-2>0\)

\(\Rightarrow x>4\)

Answer 2