Mitternachtsformel

Mitternachtsformel is the colloquial German name for the quadratic formula used to solve quadratic equations of the form ax^2 + bx + c = 0, where a ≠ 0. The solutions are given by x = (-b ± sqrt(b^2 - 4ac)) / (2a). The expression under the square root, Δ = b^2 - 4ac, is called the discriminant and determines the nature of the roots: two distinct real roots if Δ > 0, a repeated real root if Δ = 0, and two complex conjugate roots if Δ < 0. If a = 0, the equation reduces to a linear equation bx + c = 0, with the solution x = -c/b (provided b ≠ 0); if b = c = 0, every x is a solution.

The formula can be derived by completing the square on ax^2 + bx + c = 0. Dividing by a and rearranging yields (x + b/(2a))^2 = Δ/(4a^2), from which x is obtained as above.

The term Mitternachtsformel is a German-speaking educational nickname for this result, reflecting its ubiquity in school mathematics. It is widely taught in curricula across German-speaking countries and is commonly encountered in textbooks and exams as the standard method for solving quadratic equations. See also quadratic equation; discriminant; completing the square.