I've been introduced to a method of classifying conics but it's too cumbersome for me. I've discovered something that seems a little more promising on Eves' Elementary Matrix Theory:
And then,taking the rank $\rho$ of the matrix:
And then, using also the discriminant of the conic (from Aarts' Plane and Solid Geometry):
But I'm a little unsafe about using them. I mean, what could possibly go wrong against the usual approach of using rotations and translations? What would I lose by using this approach?
From what I know, the only problem that could happen is when the conic is an imaginary ellipse or hyperbola.