Question

Cho biểu thức: P\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}+\dfrac{4\sqrt{x}-3}{2\sqrt{x}-x}\right):\left(\dfrac{\sqrt{x}+2}{\sqrt{x}}-\dfrac{\sqrt{x}-4}{\sqrt{x}-2}\right)\)

a) Rút gọn P

b) Tính GTNN của \(\sqrt{P}\)

Answer 1

a)

\(P=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}+\dfrac{4\sqrt{x}-3}{2\sqrt{x}-x}\right):\left(\dfrac{\sqrt{x}+2}{\sqrt{x}}-\dfrac{\sqrt{x}-4}{\sqrt{x}-2}\right)\)

\(\Leftrightarrow P=\left(\dfrac{\sqrt{x}}{\sqrt{x}-2}+\dfrac{4\sqrt{x}-3}{\sqrt{x}\left(2-\sqrt{x}\right)}\right):\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)-\sqrt{x}\left(\sqrt{x}-4\right)}{\sqrt{x}.\left(\sqrt{x}-2\right)}\)

\(\Leftrightarrow P=\dfrac{\sqrt{x}.\sqrt{x}-\left(4\sqrt{x}-3\right)}{\sqrt{x}.\left(\sqrt{x}-2\right)}.\dfrac{\sqrt{x}.\left(\sqrt{x}-2\right)}{x-4-x+4\sqrt{x}}\)

\(\Leftrightarrow P=\dfrac{x-4\sqrt{x}+3}{4\sqrt{x}-4}\)

\(\Leftrightarrow P=\dfrac{x-3\sqrt{x}-\sqrt{x}+3}{4\left(\sqrt{x}-1\right)}\)

\(\Leftrightarrow P=\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)-\left(\sqrt{x}-3\right)}{4\left(\sqrt{x}-1\right)}\)

\(\Leftrightarrow P=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-3\right)}{4\left(\sqrt{x}-1\right)}\)

\(\Leftrightarrow P=\dfrac{\sqrt{x}-3}{4}\)

Answer 2

b) Ta có :

\(\sqrt{P}=\sqrt{\dfrac{\sqrt{x}-3}{4}}=\dfrac{\sqrt{\sqrt{x}-3}}{2}\)

vì: \(\sqrt{\sqrt{x}-3}\ge0\)

\(\Leftrightarrow\dfrac{\sqrt{\sqrt{x}-3}}{2}\ge0\)

\(\Leftrightarrow\sqrt{P}\ge0\)

dấu bằng xảy ra \(\Leftrightarrow\sqrt{\sqrt{x}-3}=0\Leftrightarrow\sqrt{x}-3=0\Leftrightarrow\sqrt{x}=3\Leftrightarrow x=9\left(TMĐK\right)\)

Vậy \(min\sqrt{P}=0khix=9\)