Question

Cho \(P=\dfrac{x\sqrt{y}+y\sqrt{x}}{\sqrt{xy}}-\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-4\sqrt{xy}}{\sqrt{x}-\sqrt{y}}-y\)

a)Rút gọn P

b)C/m P <1

Answer 1

ĐKXĐ: x≠y,x>0,y>0

a) \(P=\dfrac{x\sqrt{y}+y\sqrt{x}}{\sqrt{xy}}-\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-4\sqrt{xy}}{\sqrt{x}-\sqrt{y}}-y=\dfrac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}}-\dfrac{x+2\sqrt{xy}+y-4\sqrt{xy}}{\sqrt{x}-\sqrt{y}}-y=\sqrt{x}+\sqrt{y}-\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{x}-\sqrt{y}}-y=\sqrt{x}+\sqrt{y}-\sqrt{x}+\sqrt{y}-y=2\sqrt{y}-y\)b) Ta có \(\left(\sqrt{y}-1\right)^2>0\Leftrightarrow y-2\sqrt{y}+1>0\Leftrightarrow1>2\sqrt{y}-y\Leftrightarrow P< 1\)