Question

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

a) Tìm ĐKXĐ

b) Rút gọn P

c) Tìm x để P>0

d) Tìm P max

Answer 1

a: ĐKXĐ: x>=0; x<>1

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

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

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

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

c: Để P>0 thì \(\sqrt{x}-1< 0\)

=>0<x<1