Dr.Samer's Method for Solving General Cubic Equations - A Complex Analytic Method for Solving via Inverse Function Expansion

Dr.Samer’s method for solving general cubic equationsAuthor: H. Snober, Samer Description / AbstractThis document presents a modern method for finding the three real roots or the only real root with the other two imaginary roots of a general cubic equation whatever a, b and c values are. f(x) = x³ + ax² + bx + c = 0 The method utilizes trigonometric functions and introduces a specific discriminant Δ₃ defined as: Δ₃ = (2a³ - 9ab + 27c) / (2(a² - 3b)√(a² - 3b)) The roots x₁, x₂, and x₃ are determined using this discriminant. The work also covers special cases where a² = 3b and provides a unique geometric interpretation involvingequilateral triangles, demonstrating that the roots correspond to the vertices of such atriangle within a circle. Comparison with Cardano's MethodUnlike the traditional Cardano's method, which often involves complex numbers during intermediate steps (the casus irreducibilis) even for real roots, Dr.samer’s method provides:* Direct Real-Domain Calculation: It remains within the domain of real numbers by using trigonometric identities. * Geometric Clarity: While Cardano's approach is purely algebraic, this method links the roots to the geometry of an equilateral triangle, providing a visual confirmation of the solutions. * Unified Formula: The introduction of the Δ₃ discriminant offers a more streamlined path to finding all three roots simultaneously using a single trigonometric framework. Keywords1 - Cubic Equations2 - Analytical Geometry3 - Trigonometric Solutions4 - Roots of Polynomials