discriminant

Discriminant is a term used in mathematics to denote a quantity associated with a polynomial, a quadratic form, or more generally with an algebraic object, that encodes information about the roots or geometric properties. In many contexts, the discriminant reveals whether roots collide (multiplicity) and what the qualitative nature of the solution set looks like. It is typically invariant under certain transformations, such as permuting roots or rotating coordinates.

For a quadratic equation of the form ax^2 + bx + c = 0 with a ≠ 0, the discriminant is Δ = b^2 − 4ac. The sign and value of Δ determine the nature of the roots: Δ > 0 yields two distinct real roots, Δ = 0 yields a repeated real root, and Δ < 0 yields two complex conjugate roots.

For a general polynomial P(x) = a_n x^n + ... + a_0 with roots r_1, ..., r_n, the discriminant is Δ = a_n^{2n−2} ∏_{i<j} (r_i − r_j)^2. It vanishes exactly when the polynomial has a multiple root. This discriminant can also be computed as the resultant of P and its derivative and is a symmetric function of the roots, reflecting their separations.

In coordinate geometry, the discriminant of a second-degree curve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is Δ = B^2 − 4AC. After appropriate changes of coordinates, the sign of Δ helps classify the conic: Δ > 0 indicates a hyperbola, Δ = 0 a parabola (or a pair of parallel lines), and Δ < 0 an ellipse (or imaginary ellipse).

In number theory, discriminants extend to integers associated with polynomials or number fields, encoding arithmetic information such as ramification and the complexity of the field or form.