Quadratic

Quadratic refers to relations or expressions of the second degree. In algebra, a quadratic polynomial is a polynomial of degree two in one variable, written as ax^2 + bx + c, where a ≠ 0. A quadratic function is a function of the form f(x) = ax^2 + bx + c, whose graph is a parabola. The graph opens upward if a > 0 and downward if a < 0.

The standard form is ax^2 + bx + c. A closely related form is the vertex form: a(x − h)^2 + k, where h = −b/(2a) and k = f(h). The axis of symmetry is x = h. The y-intercept is c, and the discriminant Δ = b^2 − 4ac determines the roots: Δ > 0 yields two distinct real roots; Δ = 0 yields a repeated real root; Δ < 0 yields two complex roots.

Solutions can be found by factoring, completing the square, or using the quadratic formula x = [−b ± sqrt(Δ)]/(2a). If Δ < 0, the roots are complex conjugates. The graph is a parabola with vertex at (h, k) and can be expressed in vertex form to emphasize its turning point. The sum and product of the roots (when they exist) relate to the coefficients: sum = −b/a and product = c/a.

Applications of quadratic equations and functions appear in physics, engineering, economics, and data modeling. They describe projectile motion, area optimization, and quadratic regression. If a = 0, the expression is linear rather than quadratic. The term quadratic derives from Latin quadratus, reflecting its second-degree nature.