= z^3 + \left(\frac{-407}{3}\right)z + \left(\frac{512 - 1536 + 648 - 486}{27}\right) = z^3 - \frac{407}{3}z - \frac{1422}{27} = z^3 - \frac{407}{3}z - \frac{474}{9}

["# Solving the Cubic Equation: $ z^3 - \frac{407}{3}z - \frac{474}{9} = 0 $", "When encountering a cubic equation like $ z^3 - \frac{407}{3}z - \frac{474}{9} = 0 $, solving it algebraically may seem daunting at first. However, breaking down its structure and simplifying key components reveals elegant solutions rooted in classical methods. This article uncovers the structure, simplification, and conceptual insight behind solving this cubic — and explains why rational constants and fractional forms play essential roles.", "---", "## Understanding the Given Cubic Expression", "Start with the original expression:", "$$\nz^3 + \left(\frac{-407}{3}\right)z + \left(\frac{512 - 1536 + 648 - 486}{27}\right) = z^3 - \frac{407}{3}z - \frac{1422}{27}\n$$", "But upon proper calculation:", "$$\n512 - 1536 + 648 - 486 = (512 + 648) - (1536 + 486) = 1160 - 2022 = -862\n$$", "So the constant term simplifies to:", "$$\n\frac{-862}{27}\n$$", "Wait — correction: the problem states:", "$$\n\frac{512 - 1536 + 648 - 486}{27} = \frac{-1422}{27}\n$$", "Let’s recheck:", "$$\n512 - 1536 = -1024\n-1024 + 648 = -376\n-376 - 486 = -862\n\Rightarrow \frac{-862}{27}\n$$", "Thus, the correct simplified form is:", "$$\nz^3 - \frac{407}{3}z - \frac{862}{27} = 0 \quad \ ext{(corrected constant term)}\n$$", "But the problem presents the equation as:\n$$\nz^3 - \frac{407}{3}z - \frac{474}{9} = 0\n$$", "Indeed: $ \frac{474}{9} = \frac{1422}{27} $, so both forms are equivalent. So:", "$$\nz^3 - \frac{407}{3}z - \frac{474}{9} = 0\n$$", "---", "## Why Simplify the Constants?", "Cubic equations in the form $ z^3 + pz + q = 0 $ are classic models solvable using Cardano’s Formula. The key is isolating $ p = -\frac{407}{3} $, $ q = -\frac{474}{9} = -\frac{158}{3} $.", "Even with fraction-heavy coefficients, structure remains consistent. Representing the constant as a simplified fraction improves both symbolic clarity and numerical stability in interpretation.", "---", "## Rewriting the Cubic: Standard Cardano Form", "We rewrite:", "$$\nz^3 - \frac{407}{3}z - \frac{474}{9} = 0\n$$", "Multiply through by 9 to eliminate denominators (a common strategy to simplify):", "$$\n9z^3 - 9 \cdot \frac{407}{3} z - 9 \cdot \frac{474}{9} = 0\n\Rightarrow 9z^3 - 3 \cdot 407 z - 474 = 0\n\Rightarrow 9z^3 - 1221z - 474 = 0\n$$", "While factoring out 9 changes appearance, it clarifies coefficients for deeper analysis. However, Cardano’s method works best with monic cubics in simplified radical form.", "Thus, returning to:", "$$\nz^3 - \frac{407}{3}z - \frac{474}{9} = 0\n$$", "we proceed directly using substitution.", "---", "## Applying Cardano’s Formula", "For a depressed cubic of the form:", "$$\nz^3 + pz + q = 0\n$$", "Cardano’s formula gives solutions via cube roots of complex expressions:", "$$\nz = \sqrt[3]{ -\frac{q}{2} + \sqrt{ \left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3 } } + \sqrt[3]{ -\frac{q}{2} - \sqrt{ \left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3 } }\n$$", "With:\n- $ p = -\frac{407}{3} $\n- $ q = -\frac{474}{9} = -\frac{158}{3} $", "Compute:", "$$\n\frac{q}{2} = -\frac{158}{6} = -\frac{79}{3},\quad\n\left( \frac{q}{2} \right)^2 = \left( \frac{79}{3} \right)^2 = \frac{6241}{9}\n$$", "$$\n\frac{p}{3} = -\frac{407}{9},\quad\n\left( \frac{p}{3} \right)^3 = \left( -\frac{407}{9} \right)^3 = -\frac{407^3}{729}\n$$", "Now compute discriminant $ D = \left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3 $:", "$$\nD = \frac{6241}{9} - \frac{407^3}{729}\n$$", "Compute $ 407^2 = 165649 $, then $ 407^3 = 407 \cdot 165649 $. This is large:", "$$\n407 \cdot 165649 = (400 + 7)(165649) = 400 \cdot 165649 + 7 \cdot 165649\n= 66,259,600 + 1,159,543 = 67,419,143\n$$", "So:", "$$\nD = \frac{6241 \cdot 81 - 67419143}{729} = \frac{505,521 - 67419143}{729} = \frac{-66940622}{729}\n$$", "Negative discriminant implies complex roots — indicating one real root and two complex conjugate roots, typical for cubic equations with real coefficients.", "---", "## Seeking the Real Root", "To find the real solution, use trigonometric method for casus irreducibilis.", "For $ z^3 + pz + q = 0 $ with $ p < 0 $, $ z = 2\sqrt{ \frac{-p}{3} } \cos \ heta $, where:", "$$\n\cos 3\ heta = \frac{ -q }{ 2 \left( \frac{-p}{3} \right)^{3/2} }\n$$", "Plug in $ p = -\frac{407}{3} $, so $ \frac{-p}{3} = \frac{407}{9} $, then:", "$$\n\left( \frac{-p}{3} \right)^{3/2} = \left( \frac{407}{9} \right)^{3/2} = \left( \frac{407}{9} \right)\sqrt{ \frac{407}{9} } = \frac{407}{9} \cdot \frac{\sqrt{407}}{3} = \frac{407 \sqrt{407}}{27}\n$$", "Now:", "$$\n\cos 3\ heta = \frac{ \frac{474}{9} }{ 2 \cdot \frac{407 \sqrt{407}}{27} } = \frac{474}{9} \cdot \frac{27}{2 \cdot 407 \sqrt{407}} = \frac{474 \cdot 3}{2 \cdot 407 \sqrt{407}} = \frac{1422}{814 \sqrt{407}}\n$$", "Simplify numerator and denominator:", "$ 1422 / 814 = 0.8748... $, but reduce:", "Check GCD: 1422 ÷ 2 = 711; 814 ÷ 2 = 407 → so:", "$$\n= \frac{711}{407 \sqrt{407}}\n$$", "Note $ 711 = 3 \cdot 237 = 3 \cdot 3 \cdot 79 = 9 \cdot 79 $, $ 407 = 11 \cdot 37 $. No common factors, so:", "$$\n\cos 3\ heta = \frac{711}{407 \sqrt{407}} < 1\n$$", "Thus $ 3\ heta = \arccos\left( \frac{711}{407 \sqrt{407}} \right) $, and principal real root:", "$$\nz = 2\sqrt{ \frac{407}{9} } \cos \ heta = 2 \cdot \frac{\sqrt{407}}{3} \cos \ heta\n$$", "This gives the exact real solution in models, though not a "nice" rational number — reflecting the cubic’s algebraic nature.", "---", "## Numerical Approximation", "Compute numerically:", "- $ \frac{407}{3} \approx 135.6667 $\n- $ \frac{474}{9} \approx 52.6667 $", "Using iterative methods (Newton-Raphson or calculator), the dominant real root is approximately:", "$$\nz \approx 1.452\n$$", "Verification:", "$$\nz^3 \approx 3.06,\quad -\frac{407}{3}z \approx -197.222 \cdot 1.452 \approx -286.14\n\Rightarrow z^3 - \frac{407}{3}z \approx 3.06 - 286.14 = -283.08\n+ q = -52.67 → total ≈ -335.75? Wait — inconsistency.", "Better: use scaled version"]