What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$?
The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant equals $\operatorname{det} (A)$, where $A$ is the $3\times3$ matrix with entries $a_{11},\cdots ,a_{33}$
So the answer based on this definition, equals to $15$.
Since there is little discussion about this concept online, I find
http://zakuski.utsa.edu/~kap/Forms/shorter.pdf
where, based on his method, it gets to $60$. Which one is right?
The key point is that $$ 4 \cdot 15 = 60. $$ When Kap and I and Alexander wrote the original paper, 1997, see http://zakuski.utsa.edu/~jagy/bib.html , we separated forms with all the "off-diagonal" or $yz,zx,xy$ coefficients even. You have been given such a form. Conway has proposed calling these "integral matrix" because we get $$ \left( \begin{array}{rrr} 1 & -1 & -3 \\ -1 & -1 & 2 \\ -3 & 2 & 1 \end{array} \right) $$ which has determinant $15.$
Eventually I found that it was easier to write the many related computer programs by treating the discriminant in a uniform manner. Now i write all discriminants as in, for a well-known instance, Lehman 1992, see http://zakuski.math.utsa.edu/~kap/forms.html which is just four times the determinant.
Since you evidently have a book with an answer key, i suggest you follow that.
Lehman's discriminant is this: given $$ f(x,y,z) = a x^2 + b y^2 + c z^2 + r yz+ s zx+ t xy, $$ the discriminant is $$ \Delta = 4abc + rst - ar^2 - b s^2 - ct^2. $$ Note that this is an integer, even when $r,s,t$ are allowed to be odd numbers at times.
By the way, most authors who do quadratic forms full time would say the discriminant is $120.$ Nothing I can do about that.