quadratiques

Quadratiques, in English usually called quadratics, are polynomials of degree two or equations of degree two in one variable. A standard form is ax^2 + bx + c = 0 with a ≠ 0.

Solving relies on factoring, completing the square, or the quadratic formula x = (-b ± sqrt(b^2 - 4ac)) / (2a), where the discriminant Δ = b^2 - 4ac determines the nature of the roots. The equation has two real and distinct roots if Δ > 0, exactly one real root if Δ = 0, and two complex roots if Δ < 0.

Graphically, a quadratic defines a parabola y = ax^2 + bx + c, opening upward if a > 0 and downward if a < 0. Its axis of symmetry is x = -b/(2a), and its vertex is located at (-b/(2a), f(-b/(2a))). The graph’s shape and position are determined by the coefficients a, b, and c.

Special cases include degenerate forms: if a = 0, the equation reduces to a linear bx + c = 0; if a = b = 0, the equation is either impossible (c ≠ 0) or true for all real x (c = 0).

Quadratics appear across disciplines, from physics and engineering to economics and computer science, in contexts such as projectile motion, optimization, and curve fitting. The term quadratic derives from the Latin quadratus, meaning “square.” In French mathematical literature, the term quadratique or quadratiques is used for the same concept.