Question

Cho P = \((\dfrac{\sqrt{x}-3}{\sqrt{x}+3}+\dfrac{\sqrt{x}+3}{\sqrt{x}-3}-\dfrac{14}{9-x})\times\dfrac{\sqrt{x}-3}{2}\)

CMR \(P\ge4\)

Answer 1

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge0;x\ne9\\P=\left(\dfrac{\left(\sqrt{x}-3\right)^2+\left(\sqrt{x}+3\right)^2+14}{2\left(\sqrt{x}+3\right)}\right)\end{matrix}\right.\)

\(\Leftrightarrow P=\dfrac{x+16}{\sqrt{x}+3}\)

\(P-4=\dfrac{x-4\sqrt{x}+4}{\sqrt{x}+3}=\dfrac{\left(\sqrt{x}-2\right)^2}{\sqrt{x}+3}\ge0\)

\(P-4\ge0\Rightarrow P\ge4\) đẳng thức khi \(\sqrt{x}=2\Rightarrow x=4\) thỏa mãn đk \(\left\{{}\begin{matrix}x\ge0\\x\ne9\end{matrix}\right.\) => DPCM