Determining if a quadratic polynomial is always positive Is there a quick and systematic method to find out if a quadratic polynomial is always positive or may have positive and negative or always negative for all values of its variables?
Say, for the quadratic inequality
$$3x^{2}+8xy+5xz+2yz+7y^{2}+2z^{2}>0$$
without drawing a graph to look at its shape, how can I find out if this form is always greater than zero or has negative results or it is always negative for all non-zero values of the variables?
I tried randomly substituting values into the variables but I could never be sure if I had covered all cases.
Thanks for any help.

One of the methods when you don't know necessary and sufficient condition for the minimum of function of several variables - consider other as parameters. You know that for a function
$$
a_1x^2+b_1(y,z)x+c(y,z)
$$
the minimum is attained at $\frac{-b_1(y,z)}{2a_1}$ for $a_1>0$ and any fixed $y,z$. Then you should just substitute this into your equation and solve the minimum problem w.r.t. $y$ and then, on the third step, for $z$.
In your case: $a_1 = 3, b_1 = 8y+5z$, so you put 
$$
x = -\frac{1}{6}(8y+5z)
$$
and obtain a function
$$
\frac{1}{12}(20 y^2-56 y z-z^2)
$$
which certainly can go below zero due to the negativity of the coefficient with $z^2$.
Finally, the strict inequality never holds, since any quadratic function is equal to zero in the origin.
