Kvadratiivinen

Kvadratiivinen is a Finnish adjective used in mathematics to describe things related to a square or to second degree. It denotes objects or properties characterized by terms of the second power, and is commonly applied to polynomials, equations, and forms.

In algebra, kvadratiivinen polynomi refers to a quadratic polynomial, which has degree two. Its standard form is ax^2 + bx + c, where a ≠ 0. The roots can be found with the quadratic formula x = (-b ± sqrt(b^2 - 4ac)) / (2a). The discriminant D = b^2 - 4ac determines the nature of the roots: two distinct real roots, one real double root, or two complex roots.

Kvadratiivinen yhtälö denotes a quadratic equation, typically written as ax^2 + bx + c = 0 with a ≠ 0. The graph of a kvadratiivinen funktio, described by y = ax^2 + bx + c, is a parabola. The vertex occurs at x = -b/(2a), giving the minimum or maximum point, while the axis of symmetry is the line x = -b/(2a). The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).

In higher dimensions, kvadratiivinen muoto refers to a quadratic form, generally written as Q(x) = x^T A x, where A is a symmetric matrix. Quadratic forms are classified as positive definite, negative definite, or indefinite, with important applications in optimization, geometry, and statistics.

Etymology and usage: the term derives from Latin quadratum (square). In Finnish mathematical language, kvadratiivinen covers objects and concepts of second degree, including kvadratiivinen polynomi, kvadratiivinen yhtälö, and kvadratiivinen muoto.