- #1
Poweranimals
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Any idea how to figure this problem out?
x + 1 = 9x^3 + 9x^2
x + 1 = 9x^3 + 9x^2
Help with Quadratic Equations?
- Context: High School
- Thread starter Poweranimals
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Discussion Overview
The discussion revolves around solving the equation x + 1 = 9x^3 + 9x^2, which is identified as a cubic equation rather than a quadratic one. Participants explore various methods for solving the equation and engage in a broader conversation about the implications of Galois theory on polynomial equations.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
- Historical
Main Points Raised
- Some participants clarify that the equation is cubic and can be rewritten in standard form as 9x^3 + 9x^2 - x - 1 = 0.
- One participant suggests using Cardano's methods for solving cubic equations, while others propose simpler approaches such as substituting values for x.
- A participant demonstrates a factorization method leading to the roots x = -1, x = 1/3, and x = -1/3, arguing that complex methods are unnecessary.
- There is mention of Galois theory and its implications for polynomial equations, specifically that some equations of degree 5 and higher cannot be solved using radicals.
- Participants discuss the historical context of Galois, including his life, death, and contributions to mathematics, with varying interpretations of his motivations and the circumstances surrounding his duel.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best method to solve the cubic equation, with multiple approaches and opinions presented. The discussion about Galois theory also reveals differing interpretations of its implications.
Contextual Notes
Some participants express uncertainty about the application of Galois theory and its complexity, while others highlight the historical significance of Galois's life and contributions, indicating a mix of technical and historical discourse.
Who May Find This Useful
This discussion may be of interest to those studying algebra, particularly cubic equations and Galois theory, as well as individuals interested in the historical context of mathematical developments.
- #2
NateTG
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Poweranimals said:
First off, this belongs in a different section. General math, or homework help.
It's fairily easy to solve. What have you tried?
- #3
HallsofIvy
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In addition to NateTG's point that this is NOT a differential equation, it also is NOT a quadratic equation. It is a cubic equation that can be reduced to the standard form 9x3+ 9x2- x- 1= 0.
There exist a general formula for cubics but it is difficult to apply. Sometimes the simplest way to solve an equation is to cross your fingers and see what happens if you plug in simple numbers for x! (Hint: try negative numbers first.)
There exist a general formula for cubics but it is difficult to apply. Sometimes the simplest way to solve an equation is to cross your fingers and see what happens if you plug in simple numbers for x! (Hint: try negative numbers first.)
- #4
Dinesh
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Poweranimals said:
Hi ,
You can solve this equation using the cardon methods which can be found in the any linear algebra book
- #5
Dinesh
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Poweranimals said:
Hi ,
You can solve this equation using the cardon methods which can be found in the any linear algebra book
Dinesh
- #6
hedlund
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Poweranimals said:
This one is simple. Really simple actually...
x + 1 = 9x^3 + 9x^2
x + 1 = 9x^2(x + 1)
9x^2(x + 1) - (x + 1) = 0
(x + 1)(9x^2 - 1) = 0
x + 1 = 0 => x = -1
9x^2 - 1 = 0 => x = +- 1/3
So:
x_1 = -1
x_2 = 1/3
x_3 = -1/3
No need for cardano or anything
- #7
HallsofIvy
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Or simply try x= -1 as I originally suggested to find that
-1+1= 0= 9(-1)3+ 9(-1)2 so that x= -1 is a root. Once you know that, it is easy to see that 9x3+ 92-x-1=(x+1)(9x2-1)= (x+1)(3x-1)(3x+1).
-1+1= 0= 9(-1)3+ 9(-1)2 so that x= -1 is a root. Once you know that, it is easy to see that 9x3+ 92-x-1=(x+1)(9x2-1)= (x+1)(3x-1)(3x+1).
- #8
Ethereal
Does anyone know a link to the proof which shows that there exists no exact solution for equations of order 5 and above? Sorry I don't happen to know the exact terms for this.
- #9
humanino
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It is not going to be easy. Type Galois on google. Don't read his theory, it is too difficult. Read about his life, it is fascinating. We are very proud of Galois back in France 
To Dinesh : they are also proud of Cardan back in Italy.
To Dinesh : they are also proud of Cardan back in Italy.
- #10
matt grime
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There is a thread in this forum with galois stuff in it. titled cubic formula started by atcg, it's on the first page of the current topics somewhere.
- #11
HallsofIvy
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It's not, by the way, true that "there is no exact solution". What is true is that there exist polynomial equations with solutions that cannot be written as radicals (roots of roots of ... rational numbers).
As pointed out, Galois theory is very deep. Essentially, group theory originated as a way or simplifying the proof!
The idea:
1. A polynomial equation is "solvable by radicals" if and only if the Galois group of the polynomial is a "solvable group".
2. For any positive integer, n, there exist a polynomial of degree n whose Galois group is isomorphic to Sn, the group of permutations on n objects.
3. If n> 4, Sn is not a solvable group.
As pointed out, Galois theory is very deep. Essentially, group theory originated as a way or simplifying the proof!
The idea:
1. A polynomial equation is "solvable by radicals" if and only if the Galois group of the polynomial is a "solvable group".
2. For any positive integer, n, there exist a polynomial of degree n whose Galois group is isomorphic to Sn, the group of permutations on n objects.
3. If n> 4, Sn is not a solvable group.
- #12
humanino
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Galois died so young. He would have brought to humanity such a wider math landscape. 
- #13
vsage
IIRC from a 6th grade report on him didn't he die in a duel? tsk tsk. Different times, I suppose.
- #14
HallsofIvy
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Yes, he fought a duel with a rival over some girl who wasn't worth half of him.
There was, by the way, a suggestion that the duel was set up by the duel was "set up" by government agencies. This was during the short time right after the defeat of Napolean when the Bourbons were back on the throne. There is, in fact, no evidence for that, although Galois did spend a year in prison just before this for "threatening the life of the King".
There was, by the way, a suggestion that the duel was set up by the duel was "set up" by government agencies. This was during the short time right after the defeat of Napolean when the Bourbons were back on the throne. There is, in fact, no evidence for that, although Galois did spend a year in prison just before this for "threatening the life of the King".
- #15
humanino
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It is very much more likely that he did not care about the girl. He constantly said that he would give his life if that could trigger revolution. Apparently, he decided himself to organise the duel, and die. But the same day, other events (a general murdered) triggered the "commune" revolution. So after all, he died for nothing.
The truth is : nobody knows for sure. That is what is fascinating. The thing is : Galois was so clever, why would he stupidely die in a duel ?
The truth is : nobody knows for sure. That is what is fascinating. The thing is : Galois was so clever, why would he stupidely die in a duel ?
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