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Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field). A quadratic form in four variables is the equation of a projective surface.
For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one real root − b 2 a , {\displaystyle -{\frac {b}{2a}},} sometimes called a repeated or double root or two equal roots.
The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K ×) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative".
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
Due to its appearance in this volume, the discriminant also appears in the functional equation of the Dedekind zeta function of K, and hence in the analytic class number formula, and the Brauer–Siegel theorem. The relative discriminant of K/L is the Artin conductor of the regular representation of the Galois group of K/L.
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
The number of Americans filing new applications for unemployment benefits fell again last week, but many laid-off workers are experiencing long bouts of joblessness, keeping the door open to ...
Another instance of quadratic forms is Pell's equation =. Binary quadratic forms are closely related to ideals in quadratic fields. This allows the class number of a quadratic field to be calculated by counting the number of reduced binary quadratic forms of a given discriminant.