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June 11th, 2019, 02:44 PM   #1
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can anyone help me with this understanding of quintic equations?

this is the context:
Quote:
In the early nineteenth century, Paolo Ruffini
(1765–1822) and Niels Henrik Abel (1802–1829)11 proved that no such general
formulas utilizing the usual operations and root operations exist. This means that
there will never be a simple formula that provides the solutions for every single a, b,
c, d, e, and f in a quintic equation.
I don't get this part:
Quote:
there will never be a simple formula that provides the solutions for every single a, b,
c, d, e, and f in a quintic equation.
Can anyone help me with understanding this part of the sentence? I really need it. I don't get the writer's intention. what is it mean that solutions for every single coefficient? can you rewrite this in another word so I can understand it? I should note that I am not very familiar with this concept.
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June 11th, 2019, 02:48 PM   #2
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The general quintic is $a x^5 + b x^4 + c x^3 + d x^2 + e x + f$.

There is no general algebraic formula that determines the value of $x$ for arbitrary $a, b, c, d, e$, and $f$. By algebraic formula I mean a formula involving addition, subtraction, multiplication, division, and root extraction.

For example you may be familiar with the famous quadratic formula to solve a second degree equation. There are also formulas to solve third and fourth degree equations. But there is no general formula to solve equations of fifth degree or higher.

Perhaps the wording you quoted is a little confusing. They don't mean you are solving for the coefficients. They mean there's no formula to find $x$ given arbitrary coefficients.
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Last edited by Maschke; June 11th, 2019 at 02:52 PM.
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June 11th, 2019, 02:50 PM   #3
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Quote:
Originally Posted by amirsoly View Post
this is the context:


I don't get this part:

Can anyone help me with understanding this part of the sentence? I really need it. I don't get the writer's intention. what is it mean that solutions for every single coefficient? can you rewrite this in another word so I can understand it? I should note that I am not very familiar with this concept.
If you have the equation $\displaystyle ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0$ there is no general solution for arbitrary values of a, b, c, d, e, and f. There may be some specific values of the coefficients that can give a solution, but not generally. (Linear, quadratic, cubic, and quartic equations can all be solved using methods that involve radicals. You can't do that with quintic and above.)

Is that what you are looking for?

-Dan
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June 11th, 2019, 03:03 PM   #4
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From: iran

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When you guys explain it in this way I get it perfectly but when I read the sentence mentioned, I would be again confused. I don't get it why is written solutions for every single coefficient. if you want it to rewrite that sentence what would you say, because I think my confusion is coming from the style of writing of the writer.

Last edited by amirsoly; June 11th, 2019 at 03:13 PM.
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