Why dont we have closed form solution for algebraec equations of degree more than 2.?

Question

We can solve a quadratic equation using the equation x = (-b +/- sqrt(b^2 - 2ac)/2a. Why dont we have such closed form solution for algebraec equations of degree 3 or more? Has it not been discovered, or is it that it simply doesnt exist?

Answer 1

you forgot one possibility. That being that it does exist, but is more trouble than its worth. If a "convenient" formula is an inconvenience to use, or is otherwise too conceptually complex, perhaps it wouldnt be mentioned at your level of mathematics.

To answer your question, I dont know. I am a calculus student and I dont know of any formula to solve for a degree >=3 algebraic equation.

However, I do know techniques and algorithms that make solving them a whole lot easier and more predictable

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Pascal, below me, in response to you... you have only proven my case. Thank you for showing me those equations... I truly do appreciate it. Now I know! But, they are not convenient to use, are they? They truly are more trouble than they are worth.

Answer 2

There are such closed solution for any polynomial equation whether they are quadratic, qubic, fourth power and so on.

Just use a program called Maple and type this command

> solve(a*x^3+b*x^2+c*x+d,x);

I have done it but the solution is really long , I cant put them here.
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Answer 3

They do exist for polynomial equations of degree 2 thru 4, but not higher.

Galois proved (when he was very young) that polynomials of degree 5 and higher cannot have explicit formulae for finding roots.

Answer 4

Um... if we don't have closed-form solutions for equations of degree greater than two, then what are these?

http://planetmath.org/encyclopedia/CubicFormula.html
http://planetmath.org/encyclopedia/QuarticFormula.html

Now, we don't have any closed-form solutions for quintics and higher, this is a consequence of the Abel-Ruffini theorem. See http://en.wikipedia.org/wiki/Abel-Ruffini_theorem .

Edit re: CogitoErgoCogitoSum... True, but I don't recall stating that they are at all convenient. I just said they exist.

Answer 5

Like whitesox pronounced, there comes a ingredient the place you purely can no longer get a closed variety answer. The (3x^2)^(x^2) is incredibly troublesome to combine. yet the place did those variety of human beings come from. you have no contacts.