Quadratique

Quadratique is the French term for quadratic or relating to a degree-two polynomial. In mathematics, a quadratic object is one whose highest-degree term is of degree two. In the one-variable setting, a quadratic function is typically written as f(x) = ax^2 + bx + c with a ≠ 0.

A quadratic function graphs as a parabola opening upward if a > 0 and downward if a < 0. The standard form, f(x) = ax^2 + bx + c, can be transformed into the vertex form f(x) = a(x - h)^2 + k, where h = -b/(2a) and k = f(h). The line x = -b/(2a) is the axis of symmetry of the parabola.

Quadratic equations of the form ax^2 + bx + c = 0 can be solved by several methods, including factoring, completing the square, or using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)]/(2a). The expression under the square root, b^2 - 4ac, is called the discriminant and determines the nature of the real roots: two distinct real roots when Δ > 0, a single repeated root when Δ = 0, and no real roots when Δ < 0 (in the real-number system).

Quadratics arise in a wide range of applications, including physics, engineering, economics, and computer science, where they model trajectories, optimization problems, and relationships with squared quantities. The study of quadratics dates to ancient algebraic traditions and was formalized in modern algebra, with the quadratic formula and related techniques remaining foundational tools in mathematics.