kvadratiivsed

Kvadratiivsed refers to things related to quadratic expressions or equations, that is, polynomials or relations of degree two. In one variable, the most common example is a quadratic equation of the form ax^2 + bx + c = 0, where a ≠ 0. Its roots can be found by factoring, completing the square, or applying the quadratic formula x = (-b ± sqrt(b^2 - 4ac)) / (2a). The quantity D = b^2 - 4ac, known as the discriminant, determines the nature of the roots: two distinct real roots if D > 0, one real double root if D = 0, or two complex roots if D < 0.

The graph of a quadratic function f(x) = ax^2 + bx + c is a parabola. It is symmetric about the axis x = -b/(2a), and its vertex is at (-b/(2a), c - b^2/(4a)). If a > 0 the parabola opens upward; if a < 0 it opens downward. The zeros of the function correspond to the x-intercepts.

In several variables, kvadratiivsed can refer to quadratic forms, which are homogeneous polynomials of degree two, often written as q(x) = x^T A x with A being a square matrix. Quadratic forms arise in optimization, geometry, and physics, and they can be analyzed for properties like definiteness and eigenstructure.

Applications of kvadratiivsed include projectile motion, area and revenue optimization, and curve fitting with quadratic models. The term thus covers both single-variable quadratic equations and multivariable quadratic forms, depending on the context.