Difference between revisions of "User:Temperal/The Problem Solver's Resource4"
(→Third-degree and Quartic Formulas: cubic euqations) |
(→Third-degree and Quartic Formulas: spacing) |
||
Line 33: | Line 33: | ||
The three roots <math>r, s, t</math> of a cubic polynomial equation <math>x^3 + ax^2 + bx + c = 0</math> are given implicitly by | The three roots <math>r, s, t</math> of a cubic polynomial equation <math>x^3 + ax^2 + bx + c = 0</math> are given implicitly by | ||
− | <math>r | + | <math>r=- \frac {a}{3} + \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}+\left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math> |
− | + | <math>s=- \frac {a}{3} - \frac {1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}</math>+ \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}<math> | |
− | <math>s | + | </math>t=- \frac {a}{3} + \frac { - 1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c + \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3} - \frac {1 + i\sqrt {3}}{2} \left(\frac { - 2a^3 + 9ab - 27c - \sqrt {(2a^3 - 9ab + 27c)^2 + 4( - a^2 + 3b)^3}}{54}\right)^{1/3}$ |
− | |||
− | <math>t | ||
− | |||
Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here]. | Quartic formulas are listed [http://www.josechu.com/ecuaciones_polinomicas/cuartica_solucion.htm here]. |
Revision as of 17:25, 8 December 2007
AlgebraThis is a collection of algebra laws and definitions. Obviously, there is WAY too much to cover here, but we'll try to give a good overview. Elementary AlgebraDefinitions
Factor TheoremIff a polynomial Quadratic FormulaFor a quadratic of form Fundamental Theorems of Algebra
Third-degree and Quartic FormulasIf third-degree polynomial The three roots Quartic formulas are listed here. The general quintic equation (or an equation of even higher degree) does not have a formula. DeterminantsThe determinant of a General Formula for the DeterminantLet
Cramer's LawConsider a set of three linear equations (i.e. polynomials of degree one) Let Abstract AlgebraIncomplete. Diophantine EquationsIncomplete. |