Question

Đặt \(P=2-\dfrac{x+5}{\sqrt{x}}\times\dfrac{\sqrt{x}}{\sqrt{x}+2}\) Chứng minh \(\left|P\right|=-P\)

Answer 1

\(P=2-\dfrac{x+5}{\sqrt{x}}\cdot\dfrac{\sqrt{x}}{\sqrt{x}+2}\)

\(P=2-\dfrac{x+5}{\sqrt{x}+2}\)

\(P=\dfrac{2\left(\sqrt{x}+2\right)-x-5}{\sqrt{x}+2}\)

\(P=\dfrac{2\sqrt{x}+4-x-5}{\sqrt{x}+2}\)

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

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

Ta có: \(\sqrt{x}+2>0\forall x\)

\(\Rightarrow P=\dfrac{-\left(\sqrt{x}-1\right)^2}{\sqrt{x}+2}\le0\forall x\) (vì \(-\left(\sqrt{x}-1\right)^2\le0\forall x\) )

\(\Rightarrow\left|P\right|=-P\left(đpcm\right)\)