Cubic Equation Cardano solution

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SUMMARY

The discussion focuses on solving the cubic equation 3x³ + x² + 15x + 27 = 0 using Cardano's method. The user presents complex solutions involving imaginary numbers and square roots, specifically referencing the expressions derived from the Cardano/Tartaglia methods. The solutions include terms like 67(1 ± i√3) and involve calculations with square roots of 16305. The user expresses confidence in mastering the cubic solution through the use of online resources.

PREREQUISITES
  • Understanding of cubic equations and their properties
  • Familiarity with Cardano's method for solving cubic equations
  • Basic knowledge of complex numbers and imaginary units
  • Proficiency in algebraic manipulation and simplification
NEXT STEPS
  • Study the derivation of Cardano's formula in detail
  • Practice solving various cubic equations using Cardano's method
  • Explore the historical context and applications of Tartaglia's contributions
  • Learn about the implications of complex roots in polynomial equations
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Students studying algebra, mathematicians interested in polynomial solutions, and educators teaching advanced algebra concepts.

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Homework Statement


To solve the following equation
3x^3 + x^2 + 15x + 27 = 0

Homework Equations





The Attempt at a Solution



{(x-> 1/9{-1 - 134/(-3079+27Sqrt.(16305)^(1/2) + (-3079+27Sqrt.(16305)^(1/3)}

{x-> -1/9 + 67(1+iSqrt.3)/9(-3079+27Sqrt.(16305)^(1/2) -1/18 (1-iSqrt.3)(-3079+27Sqrt.(16305)^(1/2)}

Is this correct ?

{x-> -1/9 + 67(1-iSqrt.3)/9(-3079+27Sqrt.(16305)^(1/2) - 1/18 (1+iSqrt.3)(-3079+27Sqrt.(16305)^(1/2)}
 
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I actually mastered the cubic solution.
I used this site.
The methods of Cardano/Tartaglia are trully beautifully derived.
 

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