A First Unified Framework for Solvable Sextic Equations via Quadratic-Resolvent Rational Substitutions

We present a unified algebraic framework for constructing and solving a broad class of solvable sextic equations via general rational substitutions of the form \[ z = {x^3 + α_2 x^2 + α_1 x + α_0}{x^k + βₖ₋₁ xᵏ⁻¹ + ⋯ + β_0}, k ∈ \{1,2,3\}. \] By substituting this expression into an auxiliary quadratic equation \(z^2 + p z + q = 0\), we systematically obtain sextic equations whose coefficients are explicit polynomial functions of the substitution parameters. The framework unifies previously known constructions based on linear and quadratic denominators and extends them to a general setting with full parameter freedom. For each case \(k = 1,2,3\), explicit coefficient formulas are derived and representative subfamilies are classified. We establish a structural characterization of the generated sextics by proving that they are precisely those admitting a quadratic extension over which the polynomial factors into two cubic equations. A corresponding Galois-theoretic analysis shows that the associated Galois group is solvable and compatible with a natural \(3+3\) partition of the roots. From a computational perspective, the method reduces the sextic equation to the solution of one quadratic and two cubic equations, enabling exact solutions in radicals. Combined with a numerically stable cubic solver, the approach provides high-precision results without iterative procedures or initial guesses. The paper also introduces a degenerate solvable subfamily induced by physical sextics, embeds a real sextic equation from anisotropic elasticity (Lekhnitskii's theory), and provides explicit numerical solutions using a stable cubic solver. This demonstrates the framework's ability to bridge abstract algebraic solvability with physically derived models.