Talk:Quartic equation

message: Why is the Rational Roots therom left out? I would include it before the Galois soltion, because its essentially Proved by Galois, and its Galois simplest lemma, or at the very start when you start looking for real roots, vs Imaginary roots. Actally The article is good. Very good infact, and shows a lot of quality mathamatics. Artoftransformation 10:14, 10 November 2005 (UTC)
thread_title:  Rational Roots therom 

message: I'm currently working on an applet to calculate ferrari's solution to an equation. 
The funny thing is that the sanity check fails.(putting the found values in the equation).
I've been looking for this one whole day and I'm not sure if it is my mistake or a mistake in the formula. 
Could you please post the values you get for alpha,beta, etc... on the x^4 + 6x^2 - 60x +36 = 0 equation? This would really help me out.
:You test root finders by constructing polynomials with known answers.  Simply expand (x+a)(x+b)(x+c)(x+d) by hand or do it in the computer and feed the result into your solver.  mdf 16:46, 23 November 2006 (UTC)
thread_title:  Ferrari's solution 

message: I was wondering, anyone know what name the substitution of
: x = u - {B \over 4 A} 
is called? And how did Ferrari resolve to this method to depress the x^3 term?
: The general class of substitutions are called Tschirnhaus transformations.  As for how to arrive at these, I imagine that Ferrari probably more or less just guessed.  There is a general theory of these sorts of tricks due to Lagrange, however.  It is based on analyzing the properties of symmetric functions of the roots.  For a bit of the flavor of this, you can look at chapter 1 in Galois Theory by Stewart (its a reference at the bottom of this page). Grokmoo 19:06, 10 March 2007 (UTC)
thread_title:  Substitution x→u 

message: The first change by Mabuhelwa looks very suspicious: The signs were changed such that all S_i coincide. Furthermore the capitalisation doesn't make sense: Clashes with group S_4 and is not updated in the following text. Revert? --TilmanV 09:23, 5 June 2007 (UTC)
replies: text: Yes, I agree that the S_i should not be all equal (though I should stress that I don't completely understand the section), and I reverted the edits. Thanks for noticing this. -- Jitse Niesen (talk) 21:36, 6 June 2007 (UTC)
thread_title:  Galois theory and factorization 

message: I have removed references to Quasi-symmetric equation per Wikipedia:Articles for deletion/Quasi-symmetric equation (seems to be OR), however it would be worthwhile someone checking my edits to confirm that I haven't removed more than I needed to. 87.112.53.165 (talk) 16:08, 25 November 2007 (UTC)
thread_title:  [[Quasi-symmetric equation]] 

message: Since the quartic equation is simply the quartic function with a result of zero then it should be merged and this point emphasised. An equation has a result and a function has a return value but these are just semantics. A return value and a result are the same thing. 91.85.189.31 (talk) 19:30, 26 September 2008 (UTC)
replies: text: Good idea. Jakob.scholbach (talk) 13:23, 21 March 2008 (UTC)

text: I agree with the merge.  The function article is quite small, and could easily be incorporated into the equation one, though I think that the merged article name should be function, with equation as a redirect.  Asmeurer (talk  ♬ contribs) 23:27, 18 November 2008 (UTC)

text: I have no opposition to this. Ahead with the merge.--CarlosPatiño (talk) 19:07, 6 January 2009 (UTC)
thread_title:  Merge proposal 

message: I want to use this useful piece of mathematical technology to make some graphs. Sadly, this page lacks the critical information. Could a mathematician please add a section to explain whether one changes a, b c, d, e to make the curves flatter/steeper, change the lowest value y intercept, etc.? The article on quadratic equations does this graphically through a clever image, but a few lines of text would be great. Apologies to all mathematicians recoiling in horror at the thought of using maths.... :-)--Matt's talk 18:55, 8 January 2009 (UTC)
thread_title:  How to use these things 

message: , the above discussion from a decade ago decided on merging these two articles. Do you think it's more helpful to a reader to keep them seperate? (Please ping me as I'm not watching this page). – Thjarkur (talk) 19:54, 20 May 2020 (UTC)
thread_title:  Merge or unmerge? 

message: In the section for the solution of the degenerate case where a_4=0 in a_0x^4+a_1x^3+a_2x^2+a_3x+a_4=0  the equation is divided by x, which as mentioned can be zero (and actually in this case the root (x) that is divides by will actually be the root that equals zero. While the achieved result is correct it is mathematically wrong.
Instead of dividing a_0x^4+a_1x^3+a_2x^2+a_3x=0  by zero, the equation should be factorized x (a_0x^3+a_1x^2+a_2x+a_3)=0  and then the Zero-product property should be applied, the result will be that either x=0 , or a_0x^3+a_1x^2+a_2x+a_3=0  with the latter being a cubic equation as in the current article.
Should the article be updated to conform to correct math, or should it remain in its current state, which might be more accessible to a layman?
85.191.191.208 (talk) 01:47, 22 May 2021 (UTC)
thread_title:  Degenerate case: divide by zero 

message: After the title "Solving a depressed quartic when b≠0" the symbol "u" used in a previous equation is replaced by the symbol "x" apparently meaning the same as "u". In this context, I find it confusing, since it closes the direct path back to the solution of the (more) original equation in "x".
Why not keep using "u" from that point? Or else, why not at least write a remark indicating that symbol "u" is replaced by symbol "x"?Redav (talk) 23:35, 28 August 2022 (UTC)
replies: text: The same question is relevant for  \alpha, \beta, \gamma  which have been turned into "a", "b", "c".Redav (talk) 00:07, 29 August 2022 (UTC)
replies: text: I had already written the section for the depressed quartic on this talk page (and later deleted it from here) and I did not want to change the names of all those variables when I inserted it into the article. You can view it in the same light as the various special cases (which it really is) except that it is used by the general case. And "a", "b", "c" are much easier to write (and read) than "alpha", "beta", and "gamma". If you want to change all those variables names, I will not object provided you make no mistakes. JRSpriggs (talk) 14:09, 29 August 2022 (UTC)
replies: text: Thanks for explaining. So there was no deeper intention that I had missed, merely practical considerations, I understand. I can see how tedious a job it is. I may take up the glove (much) later, after having engaged more with a couple of demons, including the one that had me look for the solution to a quartic equation.Redav (talk) 22:06, 1 September 2022 (UTC)
thread_title:  Why replace "u" with "x"? 

message: It's neat neat how the quadratic formula is derived, but when someone wants to solve a quadratic equation, they don't re-derive it each time. They just use the formula. That's why there's a formula.
Similarly, when someone has a quartic equation to solve, it may or may not be neat that a solution can be derived in so many different ways, but it's just as unnecessary as the derivation of the of the quadratic formula. It was once, let it stay done. Give the people what that want--a formula. So here it is.
  
Given a quartic equation x^4 + a*x^3 + b*x^2 + c*x + d = 0, then define the following parameters using the coefficients a, b, c, & d:
 
K = b^2 - 3*a*c + 12*d
  
L = 2*b^3 - 9*a*b*c + 27*c^2 + 27*(a^2)*d - 72*b*d
  
M = L + sqrt(L^2 - 4*K^3)
  
N = (K/3)*(2/M)^(1/3) + (1/3)*(M/2)^(1/3)
  
P = (a^2)/4 - (2/3)*b + N
  
Then
  
x1 = -a/4 - (1/2)*sqrt(P) - (1/2)*sqrt[(a^2)/2 - (4/3)*b - N - (-a^3 + 4*a*b - 8*c)/(4*sqrt(P))]
   
x2 = -a/4 - (1/2)*sqrt(P) + (1/2)*sqrt[(a^2)/2 - (4/3)*b - N - (-a^3 + 4*a*b - 8*c)/(4*sqrt(P))]
  
x3 = -a/4 + (1/2)*sqrt(P) - (1/2)*sqrt[(a^2)/2 - (4/3)*b - N + (-a^3 + 4*a*b - 8*c)/(4*sqrt(P))]
  
x4 = -a/4 + (1/2)*sqrt(P) + (1/2)*sqrt[(a^2)/2 - (4/3)*b - N + (-a^3 + 4*a*b - 8*c)/(4*sqrt(P))]
  
Just plug in the values from your equation, and you get your answer. 
No replacing variables. No solving other equations on your way to get the answer you actually want. No if/then processes.
 
How it is derived? Take the full, expanded solution and collect terms to simplify. How was the full solution found? Don't care. Use any of the valid solutions the article already lists.  — Preceding unsigned comment added by 162.246.32.7 (talk) 02:18, 11 March 2023 (UTC)
replies: text: Nice intention, but it doesn’t work, at least not with Ferrari’s formula $x^4 + 6x^2 - 60x + 36 = 0$. CielProfond (talk) 07:43, 6 January 2024 (UTC)
thread_title:  People want a solution, not an exercise 

message: In that section it is written:
: "Using this root the term \sqrt{a + 2 y} in () is always real,...."
But there is no "()" in the whole article, which is linked.
https://en.wikipedia.org/wiki/Quartic_equation#Ferrari's_solution_in_the_special_case_of_real_coefficients 83.135.165.86 (talk) 21:20, 5 January 2024 (UTC)
thread_title:  Ferrari's solution in the special case of real coefficients