kvadratiske

Kvadratiske is the adjective used in Norwegian and related languages to describe objects related to the square of a variable or, in a broader sense, to second-degree mathematics. In mathematics, kvadratiske describes quadratic items such as equations, polynomials, polynomials in one variable, and quadratic forms. The term is commonly encountered in education, geometry, and algebra.

In its most familiar usage, kvadratiske ligninger are quadratic equations, typically written in standard form as ax^2 + bx + c = 0 with a ≠ 0. Solutions can be found by factoring, completing the square, or applying the quadratic formula x = [-b ± sqrt(b^2 − 4ac)]/(2a). The discriminant, D = b^2 − 4ac, determines the nature of the roots: two real roots if D > 0, a repeated real root if D = 0, and complex roots if D < 0. The graph of a quadratic equation in x is a parabola, opening upward when a > 0 and downward when a < 0. The vertex of the parabola lies at x = −b/(2a), and the axis of symmetry is the vertical line x = −b/(2a). The range depends on the direction of the parabola and its vertex.

Kvadratiske also describes quadratic polynomials and functions of a single variable, written as a x^2 + b x + c, and, in multivariable contexts, quadratic forms such as a x^2 + b x y + c y^2, which can be represented as v^T A v for a symmetric matrix A. These concepts extend to higher dimensions and are central in linear algebra, optimization, and geometry.