Discussion
The quadratic form \(Q(x, y, z) = 2x^2 + 6y^2 + 2z^2 + 8xz\) is:
Answer: C
Write the matrix of the form with rows \((2, 0, 4)\), \((0, 6, 0)\), \((4, 0, 2)\). The \(y\) part is separate and gives the eigenvalue \(6\). The \(x, z\) block \(\begin{pmatrix} 2 & 4 \\ 4 & 2 \end{pmatrix}\) has eigenvalues \(2 \pm 4\), that is \(6\) and \(-2\).
\[ \lambda = 6,\ 6,\ -2 \]
Eigenvalues of both signs mean the form is indefinite. It is not positive definite or semidefinite because of the negative eigenvalue \(-2\), and not negative definite because of \(6\). Check: at \(x = 1, z = -1, y = 0\) the form is \(2 + 2 - 8 = -4 < 0\), while at \(y = 1\) alone it is \(6 > 0\). The answer is option C.
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