<s>[INST] Any $(x, y, z)$ can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$? Please tell me whether there any $(x, y, z)$ 
which can satisfy the $5x^2+2y^2+6z^2-6xy-2xz+2yz<0$ ?
No process or just solve it by calculator are both fine.
Thank you.
 [/INST] We have
\begin{align}
&5x^2 + 2y^2 + 6z^2 - 6xy - 2xz + 2yz\\
= \ & 2y^2 + (-6x + 2z)y + 5x^2 - 2xz + 6z^2\\
= \ & 2\left(y + \frac{-3x+z}{2}\right)^2 - 2 \left(\frac{-3x+z}{2}\right)^2
+ 5x^2 - 2xz + 6z^2\\
= \ & 2\left(y + \frac{-3x+z}{2}\right)^2 + \frac{1}{2}x^2 + xz + \frac{11}{2}z^2 \\
= \ & 2\left(y + \frac{-3x+z}{2}\right)^2 + \frac{1}{2}(x + z)^2 - \frac{1}{2}z^2 + 
\frac{11}{2}z^2\\
= \ & 2\left(y + \frac{-3x+z}{2}\right)^2 + \frac{1}{2}(x+z)^2 + 5z^2.
\end{align}
Thus, $5x^2 + 2y^2 + 6z^2 - 6xy - 2xz + 2yz$ is never negative.
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