quadratischer

Quadratischer is the German adjective meaning quadratic, used to describe mathematical objects of degree two. In mathematics, it applies to expressions and concepts built from squares of variables, as distinct from linear (degree one) and cubic (degree three).

A quadratic equation is of the form ax^2 + bx + c = 0 with a ≠ 0. Its solutions are given by the quadratic formula x = [-b ± sqrt(b^2 - 4ac)]/(2a). The discriminant Δ = b^2 - 4ac determines the number and type of real roots. A quadratic equation can also be analyzed via factoring or completing the square.

A quadratic function f(x) = ax^2 + bx + c is represented by a parabola in the plane. The graph opens upward when a > 0 and downward when a < 0. The axis of symmetry is x = -b/(2a) and the vertex is at (−b/(2a), f(−b/(2a))). The sign of a controls concavity, and the function’s minimum or maximum occurs at the vertex.

In broader contexts, quadratic forms generalize the idea to expressions Q(x) = x^T A x with a symmetric matrix A, appearing in optimization, geometry, and physics. They describe surfaces and energy-like quantities that depend on the square of variables.

Etymology and usage: the term derives from Latin quadratus; in German, quadratisch describes functions or equations of degree two, while the English equivalent is quadratic. The concept is central in algebra and precalculus education and has historical significance in the development of algebraic solving techniques. See also quadratic equation, quadratic function, parabola, and discriminant.