1892:
research. However, rearranging a formula is really not at a very much higher level of "research" than rephrasing information into one's own words. I see no good reason at all for not including it in the article. I don't regard this as ignoring rules, because I regard it as an unreasonably pedantic interpretation of the "rules" to exclude this as "original research": it is not the sort of thing that the policy on original "research" is intended to stop. Also, as far as I can see, the formulas for the roots already quoted in the article are not referenced to any source, so if we are going to go in for the line that no formula can be quoted in the article without a source then the whole lot will have to be removed. Just stick the clearer and simpler version in, say I.
2591:
fact, that is explained in the article, just after this formula. This is the reason why
Mathematica (and also Maple) can not simplify to zero when substituting in the equation: their simplifiers make not the distinction between real and complex. I do not know which formula is provided in Mathematica, but until Maple V, the Maple's solve function provided the formula with two cubic roots and could not simplify to zero the substitution of the solution in the equation. This was corrected in version V.5 only. This shows how this error is common, even among experienced mathematicians.
31:
865:
656:
2590:
Sorry, but I disagree with the present version with a single cubic root appearing twice. The formula with two different cubic roots appears in many (maybe most) textbooks and was
Cardano's formula. It has thus to appear first. The fact that it is wrong when the square root is not real is an important
3263:
aren't taught to innocent little children: It's NOT because "when we write the roots DIRECTLY in terms of coefficients, those formulas are SOOO looong!"... but rather it's because of the complex cases and sub-cases of all sorts of special values for all discriminants involved... which in the case of
1891:
I agree with the IP editor and DVdm. This is nothing but a more concise way of writing the same thing, avoiding, for example, writing the great long square root six times. OK, if there is no reliable source that gives the formula in this precise form then it is possible to argue that it is original
397:
The historical component is indeed interesting -- one of my math professors spoke of these "duels" as rowdy public exhibitions with money and prestige on the line. If only the general populace were educated enough to enjoy such intellectual endeavors, we might have decent reality television today.
2212:
Yes. I can live with it. But I am not sure to prefer this version to the old one with the formula expanded: The new version somewhat masks the true complexity of the formula. For a beginner, and in particular for a kid, it may be interesting to know that there exist big formulas and to understand
207:
In the illustration of Omar Khayyám's geometric solution, the circle is drawn so that the parabola intersects it at the top, putting the vertical line through the center of the circle. This suggests a trivial solution, x=b/(2a^2). It would be better to draw the circle with a different diameter
1707:
667:
476:
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Interesting article. Im also interested in the historical component of this article. It speaks of
Tartaglia competing with Ferrari and Cardano and Fiore. But Im curious to know what Tartaglias method was, and how was it unique from Ferro's method?
1172:
3039:
section (since obviously his formula is faulty or incomplete). Obviously people going to the latter section (say pupils or students trying to resolve a cubic for homework) are searching for the best and more accurate expression, not for the first
2152:
1272:
1520:
2710:) is one glaringly obvious error. It has been sitting there since ages. I sent a first email to Dear Dr. Weisstein in October 2005, and four more —all friendly— emails later. Never even got a reply. They got it right
333:
860:{\displaystyle t_{k}=2{\sqrt {-{\frac {p}{3}}}}\cos \left({\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-k{\frac {2\pi }{3}}\right)-{\frac {b}{3a}}\quad {\text{for}}\quad k=0,1,2\,.}
2743:, as the link I provided. BTW, there's an apparently famous Anglo-Saxon mathematical work, an anthology, huge, full with almost every formula, but I don't know its name: maybe that might hopefully help. —
2593:
In any case, if the version with a single cubic root remains at the beginning, the whole section must to be rewritten, and I do not see how to do this, by keeping this important warning in an encyclopedic
346:
My contributions are valid and intelligent. Why do you insist on withholding valid information from the people? You do not own this page, all people are allowed to contribute meaningful information.
2051:
3550:
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651:{\displaystyle t_{k}=2{\sqrt {-{\frac {p}{3}}}}\cos \left({\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-k{\frac {2\pi }{3}}\right)\quad {\text{for}}\quad k=0,1,2\,.}
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It's the exact same thing as what's already in the article itself, only more "organized" (instead of constantly and tirelessly repeating the same lengthy expressions over and
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You don't need perfect vision to see that the angles θ are not the same, or that the three red lines intersection points on the circle do not form an equilateral triangle.
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Good catch. I modified the figure by moving the top red line slightly clockwise to make the angles match, and to make it look more like an equilateral triangle. -
1185:
2213:(without any explicit explanation) why they spent so much time on quadratic equation and not on cubic equation. This is for this kind of reasons that, with
2168:
Sorry, but the notation using a double index for two simultaneous definitions and the sophisticated use of \pm and \mp are not convenient for most readers.
3151:. At least now the article is finally complete and well-written (well-thought and well-organized: logically, as well as visually and aesthetically). —
1702:{\displaystyle u_{0}={\bar {u}}_{0}=-1\ ,\qquad u_{1}={\bar {u}}_{2}={1-i{\sqrt {3}} \over 2}\ ,\qquad u_{2}={\bar {u}}_{1}={1+i{\sqrt {3}} \over 2}}
2735:
Billions! :-) But I'm from
Romania... perhaps an Anglo-Saxon bibliography would look better. Besides, the things that Weisstein got wrong were
2419:
correct, but sourcing it seems to be a problem... Let's just hope that we can use at least one of the two (correct) links provided above... —
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Please note that a formula which is very similar to the above one appears already at the end of section "General formula for the roots.
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2154:. So I chose to "modify the modification", and instead just simplify what was already there ! Hope everyone is satisfied now. :-) —
2353:
I've found online. WTH ? This is getting really annoying, and proving to be significantly harder than I initially thought... :-\ —
2217:, many beginners ask for the solutions of the general cubic equation. The opinion of other editors would be welcome on this point.
948:
are undefined or, at least, should be taken as 1 and 0 respectively. Thus the previous formula was correct and I have restored it.
1737:
Very nice. If you have the source (book/journal/article title, author, publisher, date, page...), let's put it in the article. -
451:
Otherwise it works smoothly. (There is also a typo in the artciel on p. 349 where they say v when they mean w in one equation.)--
3201:... :D And, as you can see, those formulas are even more complex, meaning even more special cases... (evil sadistic laughter) —
2375:
might count as possibly valid links. (It's correct, alright, but I'm not so sure if it's OK to copy their content elsewhere). —
2714:, but the other junk is still sitting there. No, I don't trust anything on these sites, but some mileages seem to vary here ;-)
3131:
You beat me to it! :-) I've just now opened up my computer because I've realized that both formulas are "faulty", one when
1981:
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59:
2235:, to make people better appreciate the usefulness of the Arab ones, which they might otherwise take for granted... :-) —
3497:
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349:
And if youre going to change my wording or undo my edits, the least you could do is use proper, intelligible
English.
2626:, I think I found a way to do just that: Please take a look again at the article, and tell me if you're satisfied. —
3106:
Adding some comments on the choice or the square and cubic roots and its relation to the numbering of the solutions.
38:
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After trying to verify the following equation using my calculator and some examples, I find it is not correct.
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The denominator should be 27a^2 instead of 27a^3. please someone correct this. You can see a link here:
1167:{\displaystyle x_{i}=-{a \over 3}+u_{i}\cdot {t \over 3}+{\bar {u}}_{i}\cdot {\frac {\Delta _{0}}{3t}}}
878:
2389:
IMO, all formulas in the article were correct except the present one: Presently, you divide twice by 3
3453:
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simply do not exist: You extract a complex root of a negative number, but that's it: threre are no
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47:
17:
2147:{\displaystyle {\bar {t}}={\sqrt{\Delta _{1}+{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}} \over 2}}}
3590:
3119:
2707:
2702:. The Wolfram site is a commercial vehicle for Mathecatica and is full of errors. As an example,
2601:
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2331:
I do not understand your point: the cubic roots are 3 and -3, which produces the correct result.
2222:
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1949:
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161:
3114:, which is the most important part of the formula is visually too far from the solution itself.
2761:
better ? It looks like it's from a published book... He mentions or references a work of Viete,
2393:, inside the radicals and at the end. The resulting formula is thus homogeneous of degree -1 in
2231:
That's sort of like saying that we should intentionally write all numbers on
Knowledge (XXG) in
3552:. This could be done, but I am not sure it is a good idea: this will make less clear that when
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2929:
2918:
1893:
1485:
424:
The April 2010 issue of
Resonance has an alternative solution to a depressed cubic found here
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the cubic root (initial of "cubic", capitalized to emphasize that it is a complex expression)
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By the way, I changed {0,1,2} to {1,2,3} for consistency within the article. I made another
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It involves finding values for u,v,w, so that the cubic is reduced to (x+u)^3 = -v(x+w)^3
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1267:{\displaystyle t={\sqrt{\Delta _{1}-{\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}} \over 2}}}
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different... Apart from having the coeficients normalized, it also uses the fact that
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It's not a "calculation", let alone "research" (original or otherwise), it's simply a
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Yes, I agree. Unless someone objects, feel free to add it to the article. Ditto for
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Are you saying neither of the two trustworthy sources I offered are acceptable ? —
1819:. An abbreviation. Nothing more. Like saying "Pi" instead of "3.14...". It doesn't
1759:
again). It's not a new formula, or anything. I proposed a similar approach for the
2912:
439:
x^3 +cx + d= 0 c is -3 times q^2 and d is -2 times q^3 then you get a tautology.
46:
If you wish to start a new discussion or revive an old one, please do so on the
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one as well, also based on what was already there, only "grouped". Maths isn't
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Not to me. Dr. Math is anonymous and i.m.o. certainly does not comply with
2300:) = (0,1,0) produces the correct result. The current article scheme with (
1760:
208:(choose a different value of b) so as not to give this false impression.
3103:
Correcting some errors (for example in the reference to the discriminant)
3110:
I am not fully satisfied by the following minor fact: The definition of
2811:
The book may be copyrighted, but the link isn't. We can add it like this
2525:)/2. The current version and scheme here above on the talk page produce
3439:. The three possible values of n give 1 and the two possible values of
3144:
3028:
1789:
I agree, and it is perfectly OK, but we can't take in the article per
1726:
I have removed some blank lines from your message—hope you don't mind.
107:
There is an error in the formula for "q" in the monic trinomial form.
420:
Alternative solution to depressed published in 2010 but with a glitch
1764:
328:{\displaystyle 4\sin ^{3}(\theta )-3\sin(\theta )+\sin(3\theta )=0}
3260:
1827:, that's all (and perhaps also facilitates the understanding). —
2914:
Numerical
Recipes in Fortran 77: The Art of Scientific Computing
1861:. Make sure you mention the talk page in your edit summary. -
25:
3288:
For the current general formula, I think we could get rid of
2765:, but it's hard to locate the text within the book itself. —
2316:) = (1,0,1,0) does not. Work ahead. Where is that source? -
466:
Error in the section "Trigonometric (and hyperbolic) method"
242:. But we must associate the coefficients with the identity
3197:? :-) Because the exact same problems translate themselves
3031:
and his endeavours, I think it's best to put those in the
343:
To whoever keeps changing my edits back to the original:
118:
Or you can derive your own expression (which I did too)
3585:, the cubic roots of unit do not appear in the formula.
2721:(that thing made of paper) that list these formulae? -
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140:
The formula is correct. In your link the polynomial in
2046:{\displaystyle t\cdot {\bar {t}}={\sqrt{2}}(a^{2}-3b)}
448:
which is of no use in finding the solution (x=2 BTW).
339:
Bad
English and a bigoted insistence to CONTROL a page
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http://www.sosmath.com/algebra/factor/fac11/fac11.html
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http://www.ias.ac.in/resonance/April2010/p347-350.pdf
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Try it to solve (I have set q = 1) x^3 - 3x - 2 = 0
3054:
try to keep them as brief andf succint as this one.
1930:. I'll leave the one here for IP79 or JamesB :-) -
1797:. Bummer. Unless of course we all agree and do the
3577:
3545:{\displaystyle ({\frac {-1+i{\sqrt {3}}}{2}})^{i}}
3544:
3486:
3432:{\displaystyle ({\frac {-1+i{\sqrt {3}}}{2}})^{n}}
3431:
3374:{\displaystyle ({\frac {-1+i{\sqrt {3}}}{2}})^{n}}
3373:
3307:
2911:Press, William H.; Vetterling, William T. (1992).
2656:There must indeed have been a good reason why the
2349:was indeed incorrect... Not only that, but so was
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1793:and it clearly is not a trivial calculation as in
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2717:Now, isn't there any good proper old-fashioned
3193:, DVdm... it kinda isn't! :-| And do you know
2655:Tricky without a proper source. <irony: -->
436:Only glitch is that if in the depressed cubic
2415:opinion, they were wrong. NOW the formula is
8:
3143:, the first one becomes the second, that of
3090:Giving a more symmetric shape to the formula
156:the equation of the link before comparing.
152:as coefficient. Thus you have to divide by
3035:section of the article, as opposed to the
2660:policy was put in place :-)</irony: -->
1335:{\displaystyle \Delta _{1}=2a^{3}-9ab+27c}
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445:and you wind up with (x+1)^3 = 1(x+1)^3
103:Error in "Q" in the monic trinomial form
3272:"choices" and "discussions" there... —
3046:. Regardless of WHICH formulas you may
2903:
852:
643:
433:Just take the cube root of both sides.
2917:. Cambridge University Press. p. 179.
230:Alternative trigonometric substitution
44:Do not edit the contents of this page.
3147:, but even this one cracks down when
2545:. Tricky business without sources ;-)
7:
1388:{\displaystyle \Delta _{0}=a^{2}-3b}
1043:{\displaystyle x^{3}+ax^{2}+bx+c=0}
234:Instead of the cosine substitution
3259:, I also disagree with you on WHY
2797:But it seems copyrighted... :-( —
2695:- check that policy and then read
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3467:I guess that you mean to replace
2397:, when it should be of degree 0.
3315:and simplify a bit by replacing
977:Simpler Formulas for Cubic Roots
29:
2288:). The scheme here above with (
1823:any "new information", it just
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3086:I have edited the section for
3027:. If someone wants to mention
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2018:
1997:
1767:if you don't want it to be. —
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1592:
1544:
1477:{\displaystyle {\bar {u}}_{i}}
1462:
1127:
898:In this section, the cubic is
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2461:was not, as with my example
1966:LOL! You guys were actually
1437:{\displaystyle u_{i}^{3}=-1}
1765:"chaotic" and "complicated"
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2552:on the quartic article. -
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1954:12:42, 23 April 2013 (UTC)
1940:11:34, 23 April 2013 (UTC)
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1837:09:49, 23 April 2013 (UTC)
1811:09:36, 23 April 2013 (UTC)
1777:09:20, 23 April 2013 (UTC)
1747:07:40, 23 April 2013 (UTC)
1719:04:22, 23 April 2013 (UTC)
982:The general cubic equation
933:{\displaystyle x^{3}+px+q}
3135:, and the other one when
1731:This will not stand !! :D
958:16:50, 3 April 2013 (UTC)
891:16:04, 3 April 2013 (UTC)
870:So I try to correct it.
3215:Yet, I do like it :-) -
2783:. Looks good for me...-
2215:computer algebra systems
1444:. are the three complex
172:Geometric interpretation
166:17:01, 25 May 2012 (UTC)
134:15:08, 25 May 2012 (UTC)
3595:09:07, 9 May 2013 (UTC)
3578:{\displaystyle u_{i}=1}
3459:04:24, 9 May 2013 (UTC)
3173:In four words: nice! -
2260:Something's wrong. Try
972:23:13, 2 May 2014 (UTC)
461:20:57, 9 May 2013 (UTC)
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1825:simplifies the writing
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3487:{\displaystyle u_{i}}
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3057:That would be all. —
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1508:{\displaystyle u_{i}}
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42:of past discussions.
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3100:Removing duplicates
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18:Talk:Cubic function
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238:, we can also use
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2016:
2000:
1801:. What say we? -
1732:
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894:
877:comment added by
827:
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797:
771:
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563:
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404:comment added by
382:comment added by
355:comment added by
214:comment added by
144:is not monic, as
124:comment added by
100:
99:
54:
53:
48:current talk page
3604:
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3334:
3314:
3312:
3311:
3306:
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2927:
2908:
2459:previous version
2457:is correct. The
2153:
2151:
2150:
2145:
2143:
2141:
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2123:
2107:
2102:
2093:
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939:
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724:
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566:
564:
562:
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371:Tartaglia's work
367:
334:
332:
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263:
226:
136:
81:
56:
55:
33:
32:
26:
3612:
3611:
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3605:
3603:
3602:
3601:
3559:
3554:
3553:
3532:
3506:
3496:
3495:
3474:
3469:
3468:
3454:
3448:
3444:
3419:
3393:
3383:
3382:
3361:
3335:
3325:
3324:
3295:
3290:
3289:
3050:choose to use,
3037:General Formula
2935:
2924:
2910:
2909:
2905:
2505:=0 it produced
2455:current version
2182:Better now ? —
2079:
2078:
2055:
2054:
2021:
1980:
1979:
1970:: "my" formula
1676:
1652:
1639:
1609:
1585:
1572:
1537:
1524:
1519:
1518:
1495:
1490:
1489:
1455:
1450:
1449:
1405:
1404:
1366:
1353:
1348:
1347:
1301:
1285:
1280:
1279:
1199:
1198:
1184:
1183:
1154:
1144:
1120:
1094:
1065:
1060:
1059:
1009:
993:
988:
987:
979:
905:
900:
899:
872:
812:
786:
759:
746:
738:
735:
731:
714:
710:
671:
666:
665:
595:
568:
555:
547:
544:
540:
523:
519:
480:
475:
474:
468:
422:
399:
377:
373:
350:
341:
255:
247:
246:
232:
209:
174:
119:
105:
77:
30:
22:
21:
20:
12:
11:
5:
3610:
3608:
3600:
3599:
3598:
3597:
3574:
3571:
3566:
3562:
3539:
3535:
3529:
3523:
3518:
3515:
3512:
3509:
3503:
3481:
3477:
3462:
3461:
3442:
3426:
3422:
3416:
3410:
3405:
3402:
3399:
3396:
3390:
3368:
3364:
3358:
3352:
3347:
3344:
3341:
3338:
3332:
3302:
3298:
3286:
3285:
3284:
3254:
3253:
3252:
3251:
3250:
3249:
3248:
3247:
3246:
3245:
3244:
3166:
3165:
3164:
3163:
3108:
3107:
3104:
3101:
3098:
3091:
3084:
3083:
3082:
3081:
3080:
3079:
3078:
3077:
3076:
3075:
3074:
3073:
3072:
3071:
3070:
3069:
3059:79.113.213.225
3055:
3041:
3007:
3006:
3005:
3004:
3003:
3002:
3001:
3000:
2999:
2998:
2997:
2996:
2995:
2994:
2993:
2992:
2991:
2990:
2989:
2988:
2987:
2986:
2985:
2984:
2983:
2982:
2981:
2980:
2979:
2978:
2934:
2933:
2922:
2902:
2901:
2900:
2899:
2898:
2897:
2896:
2895:
2894:
2893:
2892:
2891:
2890:
2889:
2888:
2887:
2886:
2885:
2884:
2883:
2882:
2881:
2880:
2879:
2878:
2877:
2876:
2875:
2874:
2873:
2872:
2841:
2840:
2839:
2838:
2837:
2836:
2835:
2834:
2833:
2832:
2831:
2830:
2829:
2828:
2827:
2826:
2825:
2824:
2823:
2822:
2821:
2820:
2819:
2818:
2817:
2816:
2815:
2814:
2813:
2812:
2799:79.113.215.181
2767:79.113.213.225
2763:De emendatione
2755:
2745:79.113.213.225
2715:
2679:79.113.213.225
2628:79.113.215.181
2621:
2620:
2619:
2618:
2617:
2616:
2615:
2614:
2613:
2612:
2611:
2610:
2609:
2608:
2575:
2574:
2573:
2572:
2571:
2570:
2569:
2568:
2567:
2566:
2565:
2564:
2546:
2440:
2439:
2438:
2437:
2436:
2435:
2434:
2433:
2432:
2431:
2421:79.113.213.225
2377:79.113.213.225
2367:Unless either
2355:79.113.213.225
2258:
2257:
2256:
2255:
2254:
2253:
2252:
2251:
2250:
2249:
2248:
2247:
2237:79.113.213.225
2233:Roman numerals
2201:
2200:
2199:
2198:
2197:
2196:
2195:
2194:
2184:79.113.213.225
2156:79.113.213.225
2140:
2135:
2127:
2122:
2118:
2114:
2111:
2106:
2101:
2097:
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2086:
2082:
2074:
2068:
2065:
2042:
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2036:
2033:
2028:
2024:
2020:
2014:
2010:
2005:
1999:
1996:
1990:
1987:
1964:
1963:
1962:
1961:
1960:
1959:
1958:
1957:
1956:
1911:
1910:
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1908:
1907:
1906:
1905:
1904:
1882:
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1880:
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1846:
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1839:
1782:
1781:
1780:
1779:
1735:
1734:
1733:
1722:
1721:
1696:
1690:
1685:
1682:
1679:
1673:
1668:
1661:
1658:
1651:
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1642:
1636:
1629:
1623:
1618:
1615:
1612:
1606:
1601:
1594:
1591:
1584:
1579:
1575:
1569:
1564:
1561:
1558:
1553:
1546:
1543:
1536:
1531:
1527:
1516:
1502:
1498:
1471:
1464:
1461:
1448:of -1 , and .
1433:
1430:
1427:
1422:
1417:
1413:
1396:
1395:
1384:
1381:
1378:
1373:
1369:
1365:
1360:
1356:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1308:
1304:
1300:
1297:
1292:
1288:
1260:
1255:
1247:
1242:
1238:
1234:
1231:
1226:
1221:
1217:
1211:
1206:
1202:
1194:
1191:
1175:
1174:
1160:
1157:
1151:
1147:
1141:
1136:
1129:
1126:
1119:
1114:
1111:
1106:
1101:
1097:
1093:
1088:
1085:
1080:
1077:
1072:
1068:
1051:
1050:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1016:
1012:
1008:
1005:
1000:
996:
978:
975:
961:
960:
929:
926:
923:
920:
917:
912:
908:
868:
867:
856:
851:
848:
845:
842:
839:
836:
833:
818:
815:
811:
806:
802:
796:
792:
789:
783:
780:
776:
769:
765:
762:
752:
749:
744:
741:
734:
730:
727:
722:
719:
713:
709:
706:
699:
696:
691:
686:
683:
678:
674:
661:It should be:
659:
658:
647:
642:
639:
636:
633:
630:
627:
624:
611:
605:
601:
598:
592:
589:
585:
578:
574:
571:
561:
558:
553:
550:
543:
539:
536:
531:
528:
522:
518:
515:
508:
505:
500:
495:
492:
487:
483:
467:
464:
421:
418:
406:71.226.189.240
372:
369:
340:
337:
336:
335:
324:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
262:
258:
254:
231:
228:
206:
204:
203:
173:
170:
169:
168:
126:140.105.70.136
104:
101:
98:
97:
92:
87:
82:
75:
70:
65:
62:
52:
51:
34:
23:
15:
14:
13:
10:
9:
6:
4:
3:
2:
3609:
3596:
3592:
3588:
3572:
3569:
3564:
3560:
3537:
3527:
3521:
3516:
3513:
3510:
3507:
3479:
3475:
3466:
3465:
3464:
3463:
3460:
3457:
3451:
3445:
3424:
3414:
3408:
3403:
3400:
3397:
3394:
3366:
3356:
3350:
3345:
3342:
3339:
3336:
3322:
3318:
3300:
3296:
3287:
3283:
3279:
3275:
3274:79.113.221.50
3271:
3267:
3262:
3258:
3255:
3243:
3239:
3235:
3234:79.113.221.50
3231:
3228:
3227:
3226:
3222:
3218:
3214:
3213:
3212:
3208:
3204:
3203:79.113.221.50
3200:
3196:
3192:
3188:
3187:
3186:
3185:
3184:
3180:
3176:
3172:
3171:
3170:
3169:
3168:
3167:
3162:
3158:
3154:
3153:79.113.221.50
3150:
3146:
3142:
3138:
3134:
3130:
3129:
3128:
3127:
3126:
3125:
3121:
3117:
3113:
3105:
3102:
3099:
3096:
3092:
3089:
3088:
3087:
3068:
3064:
3060:
3056:
3053:
3049:
3045:
3042:
3040:historically.
3038:
3034:
3030:
3026:
3023:
3022:
3021:
3020:
3019:
3018:
3017:
3016:
3015:
3014:
3013:
3012:
3011:
3010:
3009:
3008:
2977:
2973:
2969:
2965:
2964:
2963:
2962:
2961:
2960:
2959:
2958:
2957:
2956:
2955:
2954:
2953:
2952:
2951:
2950:
2949:
2948:
2947:
2946:
2945:
2944:
2943:
2942:
2941:
2940:
2939:
2938:
2937:
2936:
2931:
2925:
2923:0-521-43064-X
2920:
2916:
2915:
2907:
2904:
2871:
2870:
2869:
2868:
2867:
2866:
2865:
2864:
2863:
2862:
2861:
2860:
2859:
2858:
2857:
2856:
2855:
2854:
2853:
2852:
2851:
2850:
2849:
2848:
2847:
2846:
2845:
2844:
2843:
2842:
2810:
2809:
2808:
2804:
2800:
2796:
2795:
2794:
2790:
2786:
2782:
2778:
2777:
2776:
2772:
2768:
2764:
2760:
2756:
2754:
2750:
2746:
2742:
2738:
2734:
2733:
2732:
2728:
2724:
2720:
2716:
2713:
2709:
2705:
2701:
2700:
2694:
2690:
2689:
2688:
2684:
2680:
2676:
2675:
2674:
2673:
2672:
2668:
2664:
2659:
2654:
2653:
2652:
2651:
2650:
2649:
2648:
2647:
2646:
2645:
2644:
2643:
2642:
2641:
2640:
2639:
2638:
2637:
2633:
2629:
2625:
2607:
2603:
2599:
2595:
2589:
2588:
2587:
2586:
2585:
2584:
2583:
2582:
2581:
2580:
2579:
2578:
2577:
2576:
2563:
2559:
2555:
2551:
2547:
2544:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2451:
2450:
2449:
2448:
2447:
2446:
2445:
2444:
2443:
2442:
2441:
2430:
2426:
2422:
2418:
2414:
2413:Mathematica's
2410:
2409:
2408:
2404:
2400:
2396:
2392:
2388:
2387:
2386:
2382:
2378:
2374:
2370:
2366:
2365:
2364:
2360:
2356:
2352:
2348:
2344:
2343:
2342:
2338:
2334:
2330:
2329:
2328:
2327:
2323:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2246:
2242:
2238:
2234:
2230:
2229:
2228:
2224:
2220:
2216:
2211:
2210:
2209:
2208:
2207:
2206:
2205:
2204:
2203:
2202:
2193:
2189:
2185:
2181:
2180:
2179:
2175:
2171:
2167:
2166:
2165:
2161:
2157:
2138:
2133:
2125:
2120:
2112:
2109:
2104:
2099:
2089:
2084:
2072:
2063:
2037:
2034:
2031:
2026:
2022:
2012:
2008:
2003:
1994:
1988:
1985:
1977:
1973:
1969:
1965:
1955:
1951:
1947:
1943:
1942:
1941:
1937:
1933:
1929:
1925:
1921:
1920:
1919:
1918:
1917:
1916:
1915:
1914:
1913:
1912:
1903:
1899:
1895:
1890:
1889:
1888:
1887:
1886:
1885:
1884:
1883:
1872:
1868:
1864:
1860:
1856:
1855:
1854:
1853:
1852:
1851:
1850:
1849:
1848:
1847:
1838:
1834:
1830:
1829:79.113.230.93
1826:
1822:
1818:
1814:
1813:
1812:
1808:
1804:
1800:
1796:
1792:
1788:
1787:
1786:
1785:
1784:
1783:
1778:
1774:
1770:
1769:79.113.230.93
1766:
1762:
1758:
1754:
1750:
1749:
1748:
1744:
1740:
1736:
1729:
1728:
1724:
1723:
1720:
1716:
1712:
1711:79.113.230.93
1694:
1688:
1683:
1680:
1677:
1671:
1666:
1656:
1649:
1644:
1640:
1634:
1627:
1621:
1616:
1613:
1610:
1604:
1599:
1589:
1582:
1577:
1573:
1567:
1562:
1559:
1556:
1551:
1541:
1534:
1529:
1525:
1517:
1500:
1496:
1487:
1469:
1459:
1447:
1431:
1428:
1425:
1420:
1415:
1411:
1403:
1402:
1401:
1400:
1382:
1379:
1376:
1371:
1367:
1363:
1358:
1345:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1306:
1302:
1298:
1295:
1290:
1277:
1258:
1253:
1245:
1240:
1232:
1229:
1224:
1219:
1209:
1204:
1192:
1189:
1182:
1181:
1180:
1179:
1158:
1155:
1149:
1139:
1134:
1124:
1117:
1112:
1109:
1104:
1099:
1095:
1091:
1086:
1083:
1078:
1075:
1070:
1066:
1058:
1057:
1056:
1055:
1054:has the roots
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1014:
1010:
1006:
1003:
998:
994:
986:
985:
984:
983:
976:
974:
973:
969:
965:
964:217.197.142.0
959:
955:
951:
947:
943:
927:
924:
921:
918:
915:
910:
906:
897:
896:
895:
892:
888:
884:
880:
876:
854:
849:
846:
843:
840:
837:
834:
831:
816:
813:
809:
804:
800:
794:
790:
787:
781:
778:
774:
767:
763:
760:
750:
747:
742:
739:
732:
728:
725:
720:
717:
711:
707:
704:
697:
694:
689:
684:
681:
676:
672:
664:
663:
662:
645:
640:
637:
634:
631:
628:
625:
622:
609:
603:
599:
596:
590:
587:
583:
576:
572:
569:
559:
556:
551:
548:
541:
537:
534:
529:
526:
520:
516:
513:
506:
503:
498:
493:
490:
485:
481:
473:
472:
471:
465:
463:
462:
458:
454:
449:
446:
443:
440:
437:
434:
431:
428:
427:
419:
417:
415:
411:
407:
403:
395:
393:
389:
385:
381:
368:
366:
362:
358:
357:64.134.140.53
354:
347:
344:
338:
322:
319:
313:
310:
304:
301:
298:
292:
286:
283:
280:
277:
271:
265:
260:
256:
252:
245:
244:
243:
241:
237:
229:
227:
225:
221:
217:
213:
202:
198:
194:
190:
189:
188:
187:
183:
179:
171:
167:
163:
159:
155:
151:
147:
143:
139:
138:
137:
135:
131:
127:
123:
116:
115:
111:
108:
102:
96:
93:
91:
88:
86:
83:
80:
76:
74:
71:
69:
66:
63:
61:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
3440:
3323:and writing
3320:
3316:
3269:
3256:
3194:
3190:
3149:Delta < 0
3148:
3140:
3137:Delta < 0
3136:
3132:
3111:
3109:
3094:
3085:
3051:
3047:
3043:
3036:
3032:
3024:
2913:
2906:
2780:
2740:
2736:
2718:
2698:
2623:
2622:
2592:
2550:little tweak
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2416:
2412:
2394:
2390:
2313:
2309:
2305:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2265:
2261:
2259:
1975:
1971:
1967:
1894:JamesBWatson
1824:
1820:
1816:
1756:
1752:
1398:
1397:
1343:
1275:
1177:
1176:
1053:
1052:
981:
980:
962:
945:
941:
879:Alexvong1995
873:— Preceding
869:
660:
469:
450:
447:
444:
441:
438:
435:
432:
429:
423:
400:— Preceding
396:
384:75.172.58.58
378:— Preceding
374:
351:— Preceding
348:
345:
342:
240:x = u sin(θ)
239:
236:x = u cos(θ)
235:
233:
216:64.95.214.18
210:— Preceding
205:
175:
153:
149:
145:
141:
120:— Preceding
117:
112:
109:
106:
78:
43:
37:
3264:the humble
2351:another one
2347:old formula
178:Ssscienccce
36:This is an
3266:quadratics
3048:ultimately
2741:calculated
2268:= 0 with (
1924:went ahead
1922:I already
1484:. are the
1446:cube roots
3449:Pokajanje
3199:over here
2779:From the
2453:Yes, the
1974:indeed a
453:WickerGuy
95:Archive 8
90:Archive 7
85:Archive 6
79:Archive 5
73:Archive 4
68:Archive 3
60:Archive 1
3587:D.Lazard
3257:D.Lazard
3116:D.Lazard
2966:Nice. -
2624:D.Lazard
2598:D.Lazard
2399:D.Lazard
2333:D.Lazard
2280:2) = (0,
2219:D.Lazard
2170:D.Lazard
2053:, where
1946:D.Lazard
1817:notation
950:D.Lazard
887:contribs
875:unsigned
402:unsigned
380:unsigned
353:unsigned
212:unsigned
158:D.Lazard
122:unsigned
3232:. :-)—
3189:Uhm...
3145:Cardano
3139:. When
3093:Naming
3033:History
3029:Cardano
2737:written
2509:0=-(√3+
2497:=1 and
2417:finally
2395:a,b,c,d
1928:quartic
1859:quartic
1795:wp:CALC
1761:quartic
940:. Thus
39:archive
3261:cubics
3052:please
2739:, not
2697:their
2521:2=(√3+
1799:wp:IAR
1791:wp:NOR
1346:. . .
1342:. . .
1278:. . .
1274:. . .
726:arccos
535:arccos
3319:with
3230:Quite
3141:t = 0
3133:t = 0
2706:(and
2699:about
2693:wp:RS
2658:wp:RS
2594:style
2529:0=0,
2517:1=0,
2513:)/2,
2489:with
2264:^3 +
1968:right
1488:of .
1178:where
16:<
3591:talk
3455:Talk
3381:and
3278:talk
3270:real
3238:talk
3221:talk
3217:DVdm
3207:talk
3179:talk
3175:DVdm
3157:talk
3120:talk
3063:talk
2972:talk
2968:DVdm
2919:ISBN
2803:talk
2789:talk
2785:DVdm
2771:talk
2759:this
2749:talk
2727:talk
2723:DVdm
2719:book
2712:here
2708:here
2704:here
2683:talk
2667:talk
2663:DVdm
2632:talk
2602:talk
2558:talk
2554:DVdm
2481:) =
2425:talk
2403:talk
2381:talk
2373:this
2369:this
2359:talk
2345:The
2337:talk
2322:talk
2318:DVdm
2241:talk
2223:talk
2188:talk
2174:talk
2160:talk
1950:talk
1936:talk
1932:DVdm
1898:talk
1867:talk
1863:DVdm
1833:talk
1807:talk
1803:DVdm
1773:talk
1757:over
1755:and
1753:over
1743:talk
1739:DVdm
1715:talk
1709:. —
968:talk
954:talk
944:and
883:talk
457:talk
410:talk
388:talk
361:talk
220:talk
197:talk
193:DVdm
182:talk
162:talk
148:has
130:talk
3494:by
3195:why
2757:Is
2541:2=-
2485:^3+
2473:)*(
2411:In
2371:or
1976:bit
1972:was
1926:at
1821:add
1399:and
1344:and
1276:and
826:for
705:cos
617:for
514:cos
302:sin
284:sin
257:sin
3593:)
3508:−
3446:.
3395:−
3337:−
3280:)
3240:)
3223:)
3209:)
3191:No
3181:)
3159:)
3122:)
3065:)
2974:)
2928:,
2805:)
2791:)
2773:)
2751:)
2729:)
2685:)
2669:)
2661:-
2634:)
2604:)
2596:.
2560:)
2537:,
2533:1=
2465:*(
2427:)
2405:)
2383:)
2361:)
2339:)
2324:)
2284:,-
2276:1,
2272:0,
2243:)
2225:)
2190:)
2176:)
2162:)
2117:Δ
2110:−
2096:Δ
2081:Δ
2067:¯
2032:−
1998:¯
1989:⋅
1952:)
1938:)
1900:)
1869:)
1835:)
1809:)
1775:)
1745:)
1717:)
1660:¯
1614:−
1593:¯
1560:−
1545:¯
1463:¯
1429:−
1377:−
1355:Δ
1327:27
1312:−
1287:Δ
1237:Δ
1230:−
1216:Δ
1210:−
1201:Δ
1146:Δ
1140:⋅
1128:¯
1105:⋅
1079:−
970:)
956:)
889:)
885:•
805:−
791:π
779:−
761:−
729:
708:
690:−
600:π
588:−
570:−
538:
517:
499:−
459:)
412:)
390:)
363:)
314:θ
305:
293:θ
287:
278:−
272:θ
266:
222:)
199:)
184:)
164:)
132:)
64:←
3589:(
3573:1
3570:=
3565:i
3561:u
3538:i
3534:)
3528:2
3522:3
3517:i
3514:+
3511:1
3502:(
3480:i
3476:u
3452:|
3443:i
3441:u
3425:n
3421:)
3415:2
3409:3
3404:i
3401:+
3398:1
3389:(
3367:n
3363:)
3357:2
3351:3
3346:i
3343:+
3340:1
3331:(
3321:n
3317:i
3301:i
3297:u
3276:(
3236:(
3219:(
3205:(
3177:(
3155:(
3118:(
3112:C
3095:C
3061:(
3044:2
3025:1
2970:(
2926:.
2801:(
2787:(
2769:(
2747:(
2725:(
2681:(
2665:(
2630:(
2600:(
2556:(
2543:i
2539:x
2535:i
2531:x
2527:x
2523:i
2519:x
2515:x
2511:i
2507:x
2503:d
2501:=
2499:b
2495:c
2493:=
2491:a
2487:x
2483:x
2479:i
2477:-
2475:x
2471:i
2469:+
2467:x
2463:x
2423:(
2401:(
2391:a
2379:(
2357:(
2335:(
2320:(
2314:d
2312:,
2310:c
2308:,
2306:b
2304:,
2302:a
2298:c
2296:,
2294:b
2292:,
2290:a
2286:i
2282:i
2278:x
2274:x
2270:x
2266:x
2262:x
2239:(
2221:(
2186:(
2172:(
2158:(
2139:3
2134:2
2126:3
2121:0
2113:4
2105:2
2100:1
2090:+
2085:1
2073:=
2064:t
2041:)
2038:b
2035:3
2027:2
2023:a
2019:(
2013:3
2009:2
2004:=
1995:t
1986:t
1948:(
1934:(
1896:(
1865:(
1831:(
1805:(
1771:(
1741:(
1713:(
1695:2
1689:3
1684:i
1681:+
1678:1
1672:=
1667:1
1657:u
1650:=
1645:2
1641:u
1635:,
1628:2
1622:3
1617:i
1611:1
1605:=
1600:2
1590:u
1583:=
1578:1
1574:u
1568:,
1563:1
1557:=
1552:0
1542:u
1535:=
1530:0
1526:u
1515:.
1501:i
1497:u
1470:i
1460:u
1432:1
1426:=
1421:3
1416:i
1412:u
1383:b
1380:3
1372:2
1368:a
1364:=
1359:0
1330:c
1324:+
1321:b
1318:a
1315:9
1307:3
1303:a
1299:2
1296:=
1291:1
1259:3
1254:2
1246:3
1241:0
1233:4
1225:2
1220:1
1205:1
1193:=
1190:t
1159:t
1156:3
1150:0
1135:i
1125:u
1118:+
1113:3
1110:t
1100:i
1096:u
1092:+
1087:3
1084:a
1076:=
1071:i
1067:x
1038:0
1035:=
1032:c
1029:+
1026:x
1023:b
1020:+
1015:2
1011:x
1007:a
1004:+
999:3
995:x
966:(
952:(
946:b
942:a
928:q
925:+
922:x
919:p
916:+
911:3
907:x
881:(
855:.
850:2
847:,
844:1
841:,
838:0
835:=
832:k
817:a
814:3
810:b
801:)
795:3
788:2
782:k
775:)
768:p
764:3
751:p
748:2
743:q
740:3
733:(
721:3
718:1
712:(
698:3
695:p
685:2
682:=
677:k
673:t
646:.
641:2
638:,
635:1
632:,
629:0
626:=
623:k
610:)
604:3
597:2
591:k
584:)
577:p
573:3
560:p
557:2
552:q
549:3
542:(
530:3
527:1
521:(
507:3
504:p
494:2
491:=
486:k
482:t
455:(
408:(
386:(
359:(
323:0
320:=
317:)
311:3
308:(
299:+
296:)
290:(
281:3
275:)
269:(
261:3
253:4
218:(
195:(
180:(
160:(
154:a
150:a
146:y
142:y
128:(
50:.
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