Quadratic Discriminant Solution for the Complex Roots of a Pure Cubic Equation

This paper treats a single family of cubic equations lying at the intersection of two standard topics in elementary algebra: the quadratic discriminant b^2 - 4ac and the roots of the cubic. For any real y and any nonzero real w, the difference-of-cubes factorization of w^3x^3 - y^3 yields a quadratic factor w^2x^2 + (wy)x + y^2 whose discriminant, with linear coefficient b = wy, equals -3b^2, equivalently 3b^2i^2 with i^2 = -1. When y ≠ 0, the two non-real roots of the cubic follow as scalar multiples of the primitive cube roots of unity ω = (-1 + i√3)/2 and ω̄ = (-1 - i√3)/2. The construction works in this restricted setting, the classical fact that the third cyclotomic polynomial Φ₃(x) = x^2 + x + 1 has discriminant -3 [1,10]. That fact is not new, and the contribution offered here is expository: an elementary, resolvent-free derivation that avoids the depression and resolvent steps of the general Cardano method, reduces all root extraction to a single quadratic discriminant, exhibits the imaginary unit explicitly at every stage, and makes the passage from the quadratic discriminant to the cube roots of unity visible to a reader who has met only the quadratic formula.