Solving 3rd and 4th level equations

[Gunni]

I think one of my books mentioned a way of solving third level equations (ax^3 + bx^2 + cx + d) and fourth level equations (Same as before, add nx^4) much the same way as you do with second level equations ((-B +- Sqrt(B^2 - 4AC))/2A). I have two questions, do you guys know the formulas for solving those equations and has it been proven that fifth level equations are unsolvable through a general rule?

 

[BigRedDot]

Yes, there are solutions for cubic and quartic equations, although they are cumbersome to use by hand. The Quintic does not have a general solution "by radicals," a fact that can be adduced from group theory.

http://mathworld.wolfram.com/CubicEquation.html
http://mathworld.wolfram.com/QuarticEquation.html
http://mathworld.wolfram.com/QuinticEquation.html

 

[selfAdjoint]

The story of the cubic solution is interesting.

In the early years of the 16th century an Italian professor of mathematics named "Big Scipio" - Scipione del Ferro - figured out the general solution of the cubic, a trick that had eluded mathematicians for millenia. At this time, mathematics professorships in Italy were staffed through competition. Candidates gave problems to each other to solves, and the winner of the competition got the job. So Scipione taught some of his students how to solve the cubic, as a secret, so they could beat their opponents with cubic equation problems.

It became obvious that Scipio and his students had the solution, and there was a great effort to find what it might be. A crippled man name Tartaglia figured out the method independently, and had fantasies of leaving his low-class job to become a professor. He made the mistake of boasting that he had the solution, and pretty soon a smooth operator named Girolamo Cardano came nosing around.

Cardano was a creative mathemetician but also a rogue. Almost a Shakespearean character. He weasled the solution of the cubic out of Tartaglia with a promise to keep it secret, and then turned around and published it in his next book. Tartaglia was furious, but what could he do? So the solution of the cubic equation was launched, and led to the reluctant acceptance of complex numbers.