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11470834 No.11470834 [Reply] [Original]

Why is there no formula for quintic equations and up? How do you solve them?

>> No.11470841

>>11470834
With graphing tools>>11470834

>> No.11470845

>>11470841
So there is no algebraic solution but there is a geometric solution? But isn't there an algebraic equivalent to all geometric problems as well?

>> No.11470965

>>11470845
The Abel theorem is that you can't find a general formula for degree >= with finitely many radicals. This doesn't stop you from finding an algebraic solution to a given quintic, there just isn't a general formula that works for every single one.

But also
>>11470841

>> No.11470990

There is, it's just obscenely complicated.

>> No.11471024

>>11470990
>There is

What is it then? Show me this formula.

>> No.11471047

>>11471024

OP, this anon >>11470965 is correct and the sense that you probably took this >>11470990 post in, is incorrect. If you're now imagining some finite string of symbols (tens of thousands? Millions? Nonillions?) for a GENERAL SOLUTION OF the quintic, sextic, etc USING ONLY the six operations +, -, *, /, ^, √, there is no such thing. This is precisely what Abel-Ruffini-Galois banishes. You can't come up with one single, general ALGEBRAIC FORMULA (precisely the condition of only using the above six ops with the polynomial's coefficients) for all polynomials of degree five or greater.

Logically, this does not preclude the alternatives: perhaps, in principle (notwithstanding other theorems), you might find a "general" solution for some given degree(s), but it would have to entail more exotic operations, other than the above. On the other hand, you might identify a so-called "algebraic solution", again only using the above six ops, which yet only applied as a partial, not general, solution for the given degree(s). Read carefully: both of these possibilities fall short of a general solution for a polynomial equation of a given degree, which is itself an algebraic solution as has been said.

For an example of a more exotic tool for quintics, look up Bring radicals.

>> No.11471505

>>11470834
>how do I solve a problem if I'm not getting spoonfed with a formula

>> No.11472491

>>11471024
Let f be the function x^5-x. Now, using the six operations +, -, *, /, ^, √n and f^-1 (x), we may find a general solution of the quintic, which I leave to the reader.