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[Community Question] Geometry: Factoring singular conics into linear forms

One of our user asked:

I'm looking for an easy way to factor singular conics into linear forms in order the following exercise.

Which of the following quadratic forms define a singular > conic? Write those as a product of two linear forms.

(a) $x_0^2-2x_0x_1+4x_0x_2-8x_1^2+2x_1x_2+3x_2^2$

(b) $x_0^2-2x_0x_1+x_1^2-2x_0x_2$

(c) $3x_0^2-2x_0x_1$

The matrices of (b) and (c) have full rank and so the quadratic forms are non-degenerate. For (a), we have the matrix

$$ M= \begin{bmatrix} 1 & -1 & 2 \\ -1 & -8 & 1 \\ 2 & 1 & 3 \end{bmatrix} $$

where $det(M)=0$. One way to solve the exercise would be to orthogonally diagonalize M to get rid of the mixed terms. However this involves computing the eigenvalues of M and finding the corresponding eigenvectors. So my question is: Is there a quicker way to do this or do I have to go through the calculations?


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