# Consider the quadratic form $q(x,y,z)=4x^2+y^2−z^2+4xy−2xz−yz$ over $\mathbb{R}$ then which of the following are true

Consider the quadratic form $q(x,y,z)=4x^2+y^2-z^2+4xy-2xz-yz$ over $\mathbb{R}$. Then which of the followings are true?
1.range of $q$ contains $[1,\infty)$
2.range of $q$ is contained in $[0,\infty)$
3. range=$\mathbb{R}$
4.range is contained in $[-N, ∞)$ for some large natural number $N$ depending on $q$

I am completely stuck on it. How should I solve this problem?

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Diagonalize $q$. –  Chris Eagle Dec 13 '12 at 17:22
Diagonalization would be a very lengthy process. is there any short process –  ketu Dec 13 '12 at 17:36

I think this does not require any fancy manipulation. Note that $q(0,0,z) = -z^2$ has range $(-\infty,0]$ while $q(x,0,0) = 4x^2$ has range $[0, \infty)$, so the range must be $\mathbb{R}$...
If you consider that for $x=0$ and $y=0$ we have that $q$ maps onto $(-∞,0]$ because $q(0,0,z)=-z^2$, and for $x=0$,$z=0$ we have that $q$ maps onto $[0,∞)$, then as a whole $q$ maps onto $(-∞,∞) = \mathbb{R}$.