Talk:Quartic equation

It is too long: may be is adapt for a math book not an encyclopedia

Perhaps this article would be better in a mathematical book (wikibook?). It is long, complex and difficult to read. It would be better if the article was shorter. In my opinion, all the demostration should be summarize up. A link to the full explanation of the methods could be given instead. Historic and modern view of the problem could be expand instead. AnyFile 16:57, 24 Oct 2004 (UTC)

How to solve a polynomial equation is a good example of a mathematical topic of general interest for which good information is hard to find. It's not just mathematicians who might want to know, and anyone who actually does want to know is going to want a working explanation. That being said, I think it could be made more concise, but should it be? The long part is "The General Case, along Ferrari's lines" which spends a lot of time bludgeoning the the obvious. user: Gene Ward Smith

But this is an encyclopedia not a tutorial for everything. Should I insert how to insert a PCI card in a computer or how to install a Range Hood or quote the complet text of a novel in case sombody needs it? There exist en.wikibook.org WikiBook, I suppose it could be a more appropriate place for a complete dissertation. AnyFile 12:53, 12 Nov 2004 (UTC)

I think AnyFile has a point in that long proofs and calculations do not belong in an encyclopedic article. However, I think that the final formula for the roots (however horrible it may be) should be in the article, and given the historical importance, Ferrari's approach should also be explained. Perhaps the best solution would be if "The General Case, along Ferrari's line" was made a bit more compact, if possible; a full discussion with all the gritty details minutiously explained could be given somewhere else (Wikibooks might indeed be an appropriate place for this). However, I don't think that the article is a "tutorial" or a "dissertation"; I think it is too detailed. -- Jitse Niesen 22:20, 13 Nov 2004 (UTC)

Nobody said this article is a tutorial. The article wants to describe an old, rather unique procedure for solving a quartic equation, while inserting something into something else, is quite easy (maybe you should add it to the article Greece; do not forget to mention, what happens if the thing is too long or too thick (note my subtile sense of humor. Thx)). Cheers. --83.129.175.32 02:47, 14 Nov 2004 (UTC)

I've just put in a slick solution to the quartic, since I have an actual application for solving the quartic and came here for the answer. A good first step for clearing up this article would be to eliminate all quadratic formulae from it. They are well known and can be referred elsewhere. Goatchurch 00:11, 6 November 2006 (UTC)

Very slick indeed. An upcoming article thanks you in advance. mdf 22:35, 21 November 2006 (UTC)

solved issues

Can somebody explain me, how in Quartic_equation#Reduction_to_a_biquadradic the formula $y^{4}...$ is created, please? Somehow I do not see, if it is substitution, and if yes, what by what. Thx. --Riddick 13:11, 6 Oct 2004 (UTC)

You can compute it by using a resultant, and eliminating x between

$x^{4}+cx^{2}+dx+e=0$

and y - x² - px - q = 0, where p and q are as given. User:Gene Ward Smith

---

Is it possible to predict which combination of solution for the long and short y-equations leads to a valid solution for equation (1) (in Quartic_equation#Reduction_to_a_biquadradic)? Thx. --Riddick 16:14, 6 Oct 2004 (UTC)

It would be if instead we used y = (ax+b)/(cx+d) but I couldn't get that to work, though it seems it should. User:Gene Ward Smith

---

I was just wondering how in the summary, x (the solution) gives only two values when we are solving a quartic which should have four solutions. Thanks! (p.voges@qut.edu.au)

The signs were a little bit tricky, but I think, I fixed it... --Riddick 13:11, 6 Oct 2004 (UTC)

---

• A certain Arne noted on Wikipedia:Reference desk that there seems to be a mistake in the solution given in the article. I had a short look and I think he is right. The formula for v, which solves equation (5), seems suspect. -- Jitse Niesen 16:08, 28 Sep 2004 (UTC)

It was me (Arne) :-)) --Riddick 16:42, 28 Sep 2004 (UTC)

Now it looks to me like the formula for v (after equation (5)) is correct for real v (I tried it with some example numbers).

• But: If v is a complex number, then the formula for v does not bring values that solve (5). --Riddick 18:25, 28 Sep 2004 (UTC)

With the coefficients A=1, B=0, C=6, D=-60 and E=36 the formula for x delivers just two solutions x1=~3.100 and x2=~0.6444 (if I toggle the +/- sign; but indeed toggling the sign of the first square root (where no +/- sign can be found), results in invalid values for x) with v=9.0097912 (which is approximately a real root of (5) but not the result of the given formula for v). The remaining two solutions can be found by the procedure in the last section. But that does not look so smooth... (I strongly believe, that is because v is a real number in this example; maybe I use the wrong calculator). When I use the formulas from the summary I get (with + instead of +/-) x=~2.275-~1.347i and v=~6.304-~1.562i (both values are no roots of their corresponding equations)

on Quartic Equation -- from MathWorld I have found another description of Ferrari's solution (I just tested it, and its works fine (with coefficients from Ferrari's first one): x1=~3.100, x2=~0.6444, x3=~-1.872+~3.810i and x4=~-1.872-~3.810i=x3* with y1=~20.019582457; it sounds a little bit different at some lines (e. g. they demand a real root of the cubic equation)). --Riddick 21:47, 28 Sep 2004 (UTC)

• Should I change something, although I do not know, where the mistake is? --Riddick 21:47, 28 Sep 2004 (UTC)

Arne says: Yes. --Riddick 20:38, 1 Oct 2004 (UTC)

I took two formulae from Cubic equation: $u={\sqrt[{3}]{{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}$ and $v={p \over 3u}$

and inserted them at the right(?) place in this article. --Riddick 20:00, 1 Oct 2004 (UTC)

• experimental results: the sign patterns +++ = x1, -+- = x2, --- = x3, +-+ = x4 are good sign patterns (the others are not)

Hereby I recommend further tests and/or proves. Thx. --Riddick 20:38, 1 Oct 2004 (UTC)

I'm sorry but I think there is a mistake in the formula above. Indeed according to the article "Cubic Function", the root is http://upload.wikimedia.org/math/7/2/c/72cf13d865e0dcba1543d68d4c64df3f.png There's a minus before Q/2 whereas in this article there's no minus sign. I'd just need some information about this please. —Preceding unsigned comment added by 89.85.205.191 (talk) 17:15, 9 November 2008 (UTC)

1. A=1, B=0, C=6, D=-60, E=36: check
2. A=1, B=10, C=-6, D=60, E=36: check
3. A=1, B=-28, C=294, D=-1372, E=2401: check (remark: (x-7)^4 (it is a little bit tricky due to many zeros as inverse (mult)))

Why should S be 1?

As far as I understood mathematics $S$ can be -1, too, because: $\beta \in \mathbb {R} ^{-}$ can be true and by convention ${\sqrt {x}}\in \mathbb {R} ^{+}$ with $x\in \mathbb {R} ^{+}$ is true (that's why we use a $\pm$-sign in front of a ${\sqrt {.}}$-sign sometimes)...

example:

A=1
B=-2
C=3
D=-4
E=-5
a = -3*B*B/(8*A*A) + C/A;
b = B*B*B/(8*A*A*A) - B*C/(2*A*A) + D/A;
c = -3*B*B*B*B/(256*A*A*A*A) + C*B*B/(16*A*A*A) - B*D/(4*A*A) + E/A;
P = -a*a/12 - c;
Q = -a*a*a/108 + a*c/3 - b*b/8;
R = Q/2 + sqrt(Q*Q/4+P*P*P/27);
U = exp(ln(R)/3);
y = -5*a/6 + P/(3*U) - U;
S = b/(2*sqrt(a+2*y)*sqrt(y^2+2*y*a+a^2-c))
a
       1.5
b
       -2
c
       -6.4375
P
       6.25
Q
       -3.75
R
       1.66870856171474269987
U
       1.18611509075441016287
y
       ~-.67968070291750861840
S
       ~-1.00000000000000000003
sic!

I recommend a revert asap. --83.129.237.17 02:59, 13 Nov 2004 (UTC)

It's correct it can be both 1 and -1, I was thinking (not thinking clearly) that people could select the sign of root for it to be correct. In any case, I changed it to ±1, and later rewrote some of the section without the S, since I did't like S much anyway... Κσυπ Cyp   2004年11月13日 (土) 18:45 (UTC)

I did not cross check your changes closely; I hope you will do or have done at least some tests with constructed examples, that have well known solutions (hint: keep in mind, that two polynoms of degree 4 are equal, if 5 points are equal, which helps to cross check, if the coefficients A-E are correct). --83.129.175.32 02:22, 14 Nov 2004 (UTC)

Apart from removing the S (and correcting a missing A² earlier in the article, my changes shouldn't have changed the math, only the layout of the math. I just checked with A to E being pseudorandom numbers between -2 and 2, and it found all 4 roots. Κσυπ Cyp   2004年11月14日 (日) 04:42 (UTC)

Why should we always have 4 pairwise different solutions?

With $A=1,B=-1,C=-1,D=1,E=0$ we just have 3 solutions, so that the statement "Each sign pattern gives a different root" seems to be false or misleading (because: it rises the question "different from what?"). I recommend a revert or a completely different phrase (maybe by introduction of $s,r\in \{-1,1\}$). --83.129.237.17 03:26, 13 Nov 2004 (UTC)

Hopefully it's clearer now. Κσυπ Cyp   2004年11月13日 (土) 18:45 (UTC)

I do not see that "(although one would then have gotten a biquadratic equation earlier and therefore not be using Ferrari's solution)" is true (there is at least one case where $B\neq 0\wedge D\neq 0$ (e. g. $(x-1)^{4}=x^{4}-4x^{3}+6x^{2}-4x+1$) where we have just one solution), but I do not care so much anymore. Maybe the statement "The set of values for x (4 +/- possibilities) denotes all solutions of (1) with $x\in \mathbb {C}$, which can be less than 4 for various reasons." would be more professional.

Keep in mind, please, that in Mathematics precise and true statements are quite important, because else it is not an exact science but just the common nonsense like medicine science or engineer science...

Keep in mind, please, that I did not want to push you by the phrase "asap"; maybe I was just a little bit annoyed, because somebody made not so urgent changes, that have to be cross checked, which is quite time consuming, so that I will not do any cross checking anymore, because the general quality of Wikipedia articles is not what I would expect anyway, so that I do not work here anymore. Thx. --83.129.175.32 02:22, 14 Nov 2004 (UTC)

$(x-1)^{4}=x^{4}-4x^{3}+6x^{3}-4x+1=u^{4}+0u^{2}+0u$, which is a biquadratic equation, a case mentioned just before the start of the "Ferarri's solution" paragraph. Of course, it should be clearer, and say it's the β=0 case, instead of mentioning that it transforms into a biquadratic equation in a different part of the article. Κσυπ Cyp   2004年11月14日 (日) 04:42 (UTC)

Why should there be a +/- sign in the formula for R?

Since we just need one $R$, the + suffices. I recommend a revert asap. --83.129.237.17 02:59, 13 Nov 2004 (UTC)

Think it's clearer with a ± and comment saying both roots work, since then people won't worry that they might be taking the "wrong" complex root. Κσυπ Cyp   2004年11月13日 (土) 18:45 (UTC)

Since there was no further specification for the value of ${\sqrt {.}}$ it means by convention, that we look for a value $n={\sqrt {m}}$ that fulfills $n^{2}=m$ (there r special cases for $m\in \mathbb {R} ^{+}$). Your notation might be misleading, because people might think, that square roots have a sign nowadays, or that there would be more than 4 solutions (maybe 8 or even 24... huh!? :-) )... But of course I don't care for this misleading $\pm$... --83.129.175.32 02:22, 14 Nov 2004 (UTC)

I don't mind whether it's there or not. Think it's slightly better with the ±, but it's probably clear without, too. Κσυπ Cyp   2004年11月14日 (日) 04:42 (UTC)

Why should we compactify the formula for x1-4?

Compactification is not necessarily simpler... I would like to see an explanation. --83.129.237.17 02:59, 13 Nov 2004 (UTC)

I think "Divide both sides by −4, and move the −β2/4 to the right," is simpler than "Multiply both sides by −1, then add β2 to both sides, divide both sides by 4, then subtract β2/4 from both sides," in this case, since it's one division versus one division, one multiplication, one addition and one subtraction. Perhaps it would be simpler just to say "Divide both sides by −4,". Κσυπ Cyp   2004年11月13日 (土) 18:45 (UTC)

I disagree, because: Many simple steps can be better than just a few more complicated steps.

Anyway I was talking of the formula for $x_{1,2,3,4}$. But I don't care anymore... --83.129.175.32 02:22, 14 Nov 2004 (UTC)

Sorry, didn't read the header, just assumed you were referring to something I did... My reply was a bit irrelevant... Think it would be better if the formula was there like you said. Κσυπ Cyp   2004年11月14日 (日) 04:42 (UTC)

Rational Roots therom

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)

Ferrari's solution

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)

In my opinion, an explanation is missing here as to why the cubic resolvent

$y^{3}+{\frac {5}{2}}ay^{2}+\left(2a^{2}-c\right)y+\left({a^{3} \over 2}-{ac \over 2}-{b^{2} \over 8}\right)=0.$ (4)

can be solved particularly easily, as is done later in Summary of Ferrari's method.

Could that be added? --83.135.165.116 (talk) 09:14, 4 January 2024 (UTC) .

Substitution x→u

I was wondering, anyone know what name the substitution of

$x=u-{B \over 4A}$

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)

Galois theory and factorization

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)

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)

Quasi-symmetric equation

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)

Merge proposal

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)

I suggest merging Quartic equation and Quartic function. Despite this page being larger, I feel that it would be better to merge to Quartic function because it is trivial to derive an equation from a function. Thoughts? —Celtic Minstrel (talk • contribs) 20:42, 6 December 2007 (UTC)

Good idea. Jakob.scholbach (talk) 13:23, 21 March 2008 (UTC)

The problem is that most non-mathematicians interested in the quartic will be wanting to solve the quartic equation, so I feel it may well be potentially confusing unless steps are made to automatically redirect searches for the "equation" to the "function" etc. User:Halothane —Preceding undated comment was added at 21:52, 27 July 2008 (UTC) Sorry; my talk link is here Halothane (talk) 17:08, 31 July 2008 (UTC)

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)

I agree! Merge it! —Preceding unsigned comment added by 67.160.119.18 (talk) 18:59, 31 December 2008 (UTC)

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

Error?

I noted that the program code given for the evaluation of quartics via Ferraris Method for beta=0 incorrectly calculates the sqrt() of a negative number for the quartic equation Ax^4+Bx^3+Cx^2+Dx+Ex=0 where A=1,B=0,C=0,D=0,E=4; x^4-4=0. In this case alpha=0 and gamma=4, resulting in the evaluation of a negative sqrt.

The equation x^4 - 4 = 0 has E = -4 not E = 4.

I have my doubts about whether the code needs to be included. The program seems to be a straightforward translation of the formulas in the article, so I wonder what the added value is. Furthermore, a proper implementation should probably take floating-point issues into account. -- Jitse Niesen (talk) 13:47, 8 January 2008 (UTC)

I think the code is useful - I certainly found it useful when coding up the method in c++ and java. I have also seen others refer to this code snippet. However, the shown code does contain an error. Also, the code as shown is missing the type brackets <>; eg complex<long double>. —Preceding unsigned comment added by 212.20.240.70 (talk) 15:13, 8 January 2008 (UTC)

I have removed the code:

 1) It is unsourced OR,
 2) it attempts to deal with the non-trivial issue of finite precision, which is not discussed in the article,
 3) it does so rather badly, with an arbitrarily chosen power of ten that ensures that a range of computed roots will be completely wrong.
    As such no one should use this code, except perhaps to demonstrate the problems in not properly dealing with the issue of finite precision in numerical software.

Inclusion of a code for solving the quartic equation should cite a reputable (i.e. peer-reviewed) publication addressing the issue of finite precision in solving this equation.

Lklundin (talk) 11:45, 19 January 2008 (UTC)

There are actually two different forms of the quartic equation that are "biquadratic".

The first form:

$a_{0}x^{4}+a_{2}x^{2}+a_{4}=0\,\!$

The second form:

$[b_{0}x^{2}+b_{1}x+b_{2}]^{2}=0\,\!$

This second form is often not recognized but covered by Ferrari's method where $\,\beta =0$

"Egg plant (talk) 06:03, 9 August 2008 (UTC)"

How to use these things

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)

Merge or unmerge?

Zotur, 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)

• Thjarkur, I just see that for linear, quadratic and cubic case exist separate articles for functions and equations. I also see that in 25 wikipedias exist separate articles for the quartic case too. In the first sentences of both articles will be link to another one, so readers would not be confused. --Zotur (talk) 13:28, 21 May 2020 (UTC)

  ​I've reinstated the article. There's quite a lot of overlap currently, but maybe that's not really a problem. – Thjarkur (talk) 14:37, 21 May 2020 (UTC)

Degenerate case: divide by zero

In the section for the solution of the degenerate case where $a_{4}=0$ in $a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x+a_{4}=0$ 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 $a_{0}x^{4}+a_{1}x^{3}+a_{2}x^{2}+a_{3}x=0$ by zero, the equation should be factorized $x(a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3})=0$ and then the Zero-product property should be applied, the result will be that either $x=0$, or $a_{0}x^{3}+a_{1}x^{2}+a_{2}x+a_{3}=0$ 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)

Why replace "u" with "x"?

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)

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)

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)

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)

People want a solution, not an exercise

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)

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)

Ferrari's solution in the special case of real coefficients

In that section it is written:

"Using this root the term ${\sqrt {a+2y}}$ in (8) is always real,...."

But there is no "(8)" 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)