This paper presents a numerically stable analytical framework for solving solvable quintic equations of the form x⁵ + 5 x³ + 5 x² + x + = 0 under the solvability condition = ²/2. The method introduces scale-invariant parameters (R, K, L) that prevent overflow/underflow and reduce the problem to a cubic resolvent. The cubic resolvent is solved using backward-stable formulas developed in previous works on cubic equations. The five roots are recovered using fifth roots of unity. Extensive numerical examples, including the King quintic and near-degenerate cases, demonstrate relative errors below 10^-13. Practical applications in quantum mechanics, robotics, control theory, and computer graphics are discussed. The algorithm is deterministic, requires no initial guess, and avoids catastrophic cancellation.
Waleed mohamed khalaf Moqadem (Sat,) studied this question.