\(4\left(xy+yz+xz\right)+x+y+z=9\)
Mặt khác ta có \(\left(x+y+z\right)^2\ge3\left(xy+yz+xz\right)\Rightarrow xy+yz+xz\le\dfrac{1}{3}\left(x+y+z\right)^2\)
\(\Rightarrow\dfrac{4}{3}\left(x+y+z\right)^2+\left(x+y+z\right)\ge9\)
\(\Leftrightarrow\left[2\left(x+y+z\right)+\dfrac{3}{4}\right]^2\ge\dfrac{441}{16}\)
\(\Leftrightarrow\left[{}\begin{matrix}2\left(x+y+z\right)+\dfrac{3}{4}\ge\dfrac{21}{4}\\2\left(x+y+z\right)+\dfrac{3}{4}\le\dfrac{-21}{4}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x+y+z\ge\dfrac{9}{4}\\x+y+z\le-3\end{matrix}\right.\) \(\Rightarrow\left(x+y+z\right)^2\ge\dfrac{81}{16}\)
Mà \(P=x^2+y^2+z^2\ge\dfrac{\left(x+y+z\right)^2}{3}\ge\dfrac{81}{16.3}=\dfrac{27}{16}\)
\(\Rightarrow P_{min}=\dfrac{27}{16}\) khi \(x=y=z=\dfrac{3}{4}\)
