\(x+y+z=4-\sqrt{xyz}\)
\(\Leftrightarrow x+y+z+\sqrt{xyz}=4\)
\(\Leftrightarrow4\left(x+y+z\right)+4\sqrt{xyz}=16\)
Ta có: \(x\left(4-y\right)\left(4-z\right)\)
\(=x\left[16-4\left(y+z\right)+yz\right]\)
\(=x\left[4\left(x+y+z\right)+4\sqrt{xyz}-4\left(y+z\right)+yz\right]\)
\(=x\left(4x+4\sqrt{xyz}+yz\right)\)
\(=x\left(2\sqrt{x}+\sqrt{yz}\right)^2\)
\(\Rightarrow\sqrt{x\left(4-y\right)\left(4-z\right)}=\sqrt{x}\left(2\sqrt{x}+\sqrt{yz}\right)\)
\(=2x+\sqrt{xyz}\)
Tương tự: \(\sqrt{y\left(4-z\right)\left(4-z\right)}=2y+\sqrt{xyz}\)
\(\sqrt{z\left(4-x\right)\left(4-y\right)}=2z+\sqrt{xyz}\)
Cộng vế theo vế các đẳng thức vừa chứng minh ta được:
\(P=2\left(x+y+z\right)+3\sqrt{xyz}=2\left(4-\sqrt{xyz}\right)+3\sqrt{xyz}=8+\sqrt{xyz}\)
