Symmetric Matrices & Quadratic Forms · Superchallenge problem

On the 9-dimensional space of all real $3\times 3$ matrices $X$ consider the quadratic form $Q(X) = \operatorname{tr}(X^2) - (\operatorname{tr}X)^2$. How many negative eigenvalues does the symmetric matrix of $Q$ (with respect to the nine entries of $X$) have?