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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 $\mathbb{R}$...

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I'm wondering if checking to see if it is positive definite will also help find us the range? – diimension Dec 13 '12 at 19:59
@diimension Perhaps there is a std methodology for quadratic forms, but if you can argue in 1 line from first principles, I always find it more intuitive. – gt6989b Dec 13 '12 at 20:11

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 $(-∞,∞) = \mathbb{R}$.

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For some basic information about writing math at this site see e.g. here, here, here and here. – Julian Kuelshammer Dec 13 '12 at 17:57

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