\(\left\{{}\begin{matrix}x+y+z=0\\xy+yz+zx=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(x+y+z\right)^2=0\\2\left(xy+yz+zx\right)=0\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x^2+y^2+z^2+2xy+2yz+2zx=0\\2xy+2yz+2zx=0\end{matrix}\right.\)
\(\Rightarrow x^2+y^2+z^2+2xy+2yz+2zx-2xy-2yz-2zx=0\)
\(\Rightarrow x^2+y^2+z^2=0\)
\(\left\{{}\begin{matrix}x^2\ge0\forall x\\y^2\ge0\forall y\\z^2\ge0\forall z\end{matrix}\right.\)
Nên: \(x^2+y^2+z^2\ge0\)
Dấu "=" xảy ra khi:
\(\left\{{}\begin{matrix}x^2=0\\y^2=0\\z^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=0\\y=0\\z=0\end{matrix}\right.\)
Vậy \(x=y=z=0\)
Ta có điều phải chứng minh
