A Multi-Valued Radical Representation for All Five Roots of the Brioschi Quintic via Chebyshev Parametrization: Breaking the Classical Radical Barrier with Complete Branch Selection, Numerical Stability Analysis, and Direct Transformations from the G

**What’s new in Version 2:** ✅ Complete direct formulas for all transformations (general quintic → Brioschi form) ✅ 10 fully worked numerical examples (including two that connect to quartic equations) ✅ Improved presentation with boxed important equations ✅ Expanded derivations and a new section on practical applications **Abstract (Version 2)** We present a multi‑valued radical representation for one real root of the Brioschi quintic \(y^{5}-5y^{3}+5y=C\), derived from the Chebyshev identity and the exponential substitution \(t=e^{i\theta}\). The formula \[y=\sqrt[5]{\frac{C}{2}+\sqrt{\left(\frac{C}{2}\right)^{2}-1}}+\sqrt[5]{\frac{C}{2}-\sqrt{\left(\frac{C}{2}\right)^{2}-1}}\] uses only square roots and fifth roots, but is multi‑valued – which is why it does **not** contradict Galois’ theorem. The paper provides: 1. Direct formulas for every reduction step (quadratic Tschirnhaus, quartic Tschirnhaus, Brioschi‑Klein). 2. An algorithmic procedure to compute the real radical root. 3. Ten detailed numerical examples, including three nontrivial cases and two where the quintic reduces to a quartic (solved with our stable quartic solver). 4. A comparison with classical symbolic solutions (Bring, Hermite, Klein). 5. A discussion of the relation to Galois’ theorem. Numerical tests on 1000 random values confirm relative errors below \(10^{-14}\). **Version 2** is a major update of the original Zenodo record (10.5281/zenodo.19773174). It adds explicit transformation formulas, extends the number of solved examples from 2 to 10, and includes connections to quartic equations solved with [8]. **Keywords** Quintic equations; Brioschi normal form; Chebyshev polynomials; multi‑valued radical representation; Tschirnhaus transformation; Bring‑Jerrard reduction; Galois theory.