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1. Discriminant

   Linked via "quadratic case"

   $$\Delta = B^2 - 4AC$$
   This $\Delta$ dictates the intrinsic type of the conic section, provided the equation represents a non-degenerate curve (i.e., the determinant of the associated $3 \times 3$ matrix involving all coefficients is nonzero) [2, 3]. The classification mirrors the quadratic case:
   If $\Delta < 0$, the curve is an ellipse or a circle.

2. Discriminant

   Linked via "quadratic case"

   $$\Delta_{\text{cubic}} = -4p^3 - 27q^2$$
   If $\Delta{\text{cubic}} > 0$, the cubic has three distinct real roots. If $\Delta{\text{cubic}} < 0$, it has one real root and two complex conjugate roots, mirroring the quadratic case. If $\Delta_{\text{cubic}} = 0$, there are multiple roots.
   Relation to Resultants