\(2xy+2x-5z=0\Rightarrow5z=2xy+2x\Rightarrow z=\frac{2}{5}xy+\frac{2}{5}x\)
\(A=x^2+2y^2+2xy+\frac{8}{5}y+z+2\)
\(A=x^2+2y^2+2xy+\frac{8}{5}y+\frac{2}{5}xy+\frac{2}{5}x+2\)
\(A=x^2+2y^2+\frac{12}{5}xy+\frac{2}{5}x+\frac{8}{5}y+2\)
\(A=x^2+\left(\frac{6y}{5}\right)^2+\left(\frac{1}{5}\right)^2+2.\frac{6}{5}xy+\frac{2}{5}x+\frac{12y}{25}+\frac{14}{25}y^2+\frac{28y}{25}+\frac{14}{25}+\frac{7}{5}\)
\(A=\left(x+\frac{6y}{5}+\frac{1}{5}\right)^2+\frac{14}{25}\left(y+1\right)^2+\frac{7}{5}\ge\frac{7}{5}\)
\(\Rightarrow A_{min}=\frac{7}{5}\) khi \(\left\{{}\begin{matrix}x=1\\y=-1\\z=0\end{matrix}\right.\)
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