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altenative solution, and in fact I came with one, which I do not publish, however
Tartaglia condition's, are unavoidable, so is the same so this seem to me like the only way, otherwise there are, numerical methods, to do it, but !!JUST AN ALGEBRAIC GENERAL SOLUTION!!, I try what you have said but always find me, in the same cubic, you can not solve a cubic equation, nor any equation solving something in a minor degree.
3474:, did a FullSimplify command, and didn't get zero. I have now checked this computation for a third time, still not getting zero. The only possibility I can think of that might justify this formula being true and Mathematica not evaluating my calculations to zero is if the principal roots are defined differently, but I would have expected that to result in a permutation of the three formulas for
2296:= 0 and DELTA < 0 but at the beginning of the article there's a separate case for DELTA = 0. I think this last case should be treated apart or at least put in the DELTA<0 case, where we have 3 real solutions . For instance when we have y = x^3, DELTA=0 and we get 3 coincident solutions that are real. Is it just that with DELTA = 0 in the second case we may get a NaN calculating theta ?
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functions or "Cubic
Equation" authorized by E.W. Weisstein too. That is, in my opinion, reason more to ask author’s permission for WP article to be completed on the basis of his works. In such case somehow could be included the determinant(s) and Bisection method of graphical resolving as per my Talk: Roots of 4X^3 = ±3X + T are equal to X-coordinates of the points where
4017:(x) applying Euler’s formulae for triple argument – came to the roots in this notation that I still consider as most suitable. Assembling the variables into the determinants could be considered as original research from the point of view of an administrator but in my opinion that is well done homework only being very useful as mnemonic for the students.
4225:. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. I searched the page history, and found 8 edits by Jagged 85 in
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decisions: one may either change many formulas or precise, when needed, that the formula is correct only in the case of a single real root. This is meaningful as
Cardano itself was unable to solve the totally real case which implies to work with non real numbers. Note that a similar correction is probably needed in the page of
3303:{\displaystyle {\begin{aligned}x_{1}=&-{\frac {a}{3}}\\&-{\frac {1}{3}}{\sqrt{\frac {2a^{3}-9ab+27c+{\sqrt {\left(2a^{3}-9ab+27c\right)^{2}-4\left(a^{2}-3b\right)^{3}}}}{2}}}\\&-{\frac {1}{3}}{\sqrt{\frac {2a^{3}-9ab+27c-{\sqrt {\left(2a^{3}-9ab+27c\right)^{2}-4\left(a^{2}-3b\right)^{3}}}}{2}}}\\\end{aligned}}}
1763:
publish his results, in which
Tartaglia did nothing. Cardano even published a book about arithmetic in the meantime and sent Tartaglia a copy to show that he kept his promise. That, and the fact that he could've ignored giving Tartaglia any credit for an independent proof should be something worthwhile to say.
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The introduction to this article is horribly written. A first glance at it does not even provide a remote definition of "cubic". And the definition of "polynomial" is also placed into question. It is inundated with a variety of links spinning elsewhere to other pages, seemingly comprehensible only to
2262:
My friend from above, there is only one and only one way to solve a cubic equation, algebraically, which was given by italian's mathematicians, in then XV century, "those men's where the choseen ones, which in their hard lives had the pleasure to solve it, as a gift of destiny" I work hard to find an
1762:
s work: Cardano published del Ferro's work. Not only that, but part of the agreement was that even if he did publish something on cubics, that he give
Tartaglia a decent amount of time to at least publish his results. And I don't believe it was just a few years, he gave Tartaglia about a decade to
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cannot be solved using earlier compass and straightedge constructions. Such a proof was impossible with the technology of that time, as it needs the notion of the degree of a field extension. By the way, the history section should mention the cubic equations which were studied by the ancient Greeks,
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I've had another looka t what you've said above. The links look okay. Are you saying that what you ave written this time is well sourced rather than your own work? That you said what you put in before was your own original research was the reason it was removed but well cited work rephrasing in your
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Another inadequacy at WP article is a choice of the functions for both graphs being, due to Q*q = 0 = T, marginal case (becoming quadratic 4X^2 – 3 = 0) that as such may not be promoted as representative ones. One of them could be replaced with a graph designed by means of
Graphic explorer for cubic
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Yes, I agree that is a little odd. This was rather a hard decision. Switch it around if you like, but it would require a little fiddling with the intro section (ie before the table of contents). The reason I chose "function" instead of "equation" is that it was easier (for me) to derive the equation
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Another way to force (at least numeric) correct simplification to zero in Maple is by not using root(X,3), but surd(X,3), which gives the root closest to X, as opposed to the principal root. For instance, calculate root(-8.,3) in Maple (or (-8.)^(1/3) in
Mathematica). You don't get -2. In Maple the
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There is one in "Nature of the Roots" and another in "Root-Finding
Formula". I think they differ by a multiplicative factor of -1/(108 a^4). A consequence is that in one case, there are 3 real roots if it is positive, but in the other case there are 3 real roots if it is negative. Shouldn't they
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Radical form (78) is identical to 4X^3 – 3PX = (q^2/p^3)^(1/2) = T ≥ 0 in my notation where, due to unary operators Q = 0; ±1 and P = sign(b^2 – 3ac) = ±1, coefficients at (78) and sign ambiguity at (83) are avoided and +k*120°is added after C at last line. Both variables are given in terms of one
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Could you explain what makes it critical information? I agree that applied math has its importance; I just don't see what makes this particular use of a graph useful. Also, the reason that such a thing is not explored for cubics is it would be considerably more complicated than quadratics as there
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There is one source that does say that
Tartaglia broke his promise, that is Mathematics: From the Birth of Numbers by Jan Gullberg, page 316. It is prudent to say there have been several areas in this book that have been inaccurate (still an excellent book), so I am not saying you are wrong, I am
4373:
The bit which really looks problematic to me is the way the
Egyptians are described right at the beginning of the history section. The text messes them up with the Greeks. I haven't any access to the source to see what could be said reasonably, The bit about Omar Khayyám and Sharaf al-Dīn al-Tūsī
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I'm glad that “you've another look”. A month ago I said “as far as I know it is my original work” but now it is clear to me that L.E. Dickson revealed essentially same formulae at 1914. In algebra the truth is must: formula either does or doesn’t work but ways to get same one can be different. In
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is not allowed on wikipedia. The removal of your contribution was correct. This talk page is for discussion about improving the article, it is not a personal notepad. I have removed the original research and email above. Whether the original research i correct or has advantages is not the point.
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A significant problem it seems to me is that at the introduction (top of page) we have the usual a,b,c,d form of the function. However, at the begining of the Cardano part, we start with a monic form (thats OK) BUT with coefficients 1,a,b,c !!! this is a really bad idea - can we please keep the
4323:
In the contributions of Jagged 85 to this article, there is not only misuse of sources but also historical assertions which are clearly false: I read: "He discovered that a cubic equation cannot be solved using earlier compass and straightedge constructions". This means that he has shown that
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Some preceding comments have noticed inconsistencies in the formulas. It seems that this comes from the imprecision of the definition of cubic roots. I have corrected this in the general formula but not in the remaining of the article. This has yet to be done. This needs some work, and several
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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 or c to make the curves flatter/steeper, change the y intercept, etc.? The article on
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are four degrees of freedom instead of just three. This can be easily demonstrated when one considers that a quadratic can be written easily in the form a(x-h)^2+k so that each translation or stretch factor is isolated; to do this for cubics would be horrendously ugly. --Fusionshrimp
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0 can’t be obtained by means of any of the formulae given at WP article that is therefore uncompleted. In order to overcome same problem L.E. Dickson almost a century ago commencing from Depressed cubic (15) and setting substitution (75) established regular Radical form of cubic
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I wasn't aware of the function pages; I only came across this one when I saw the suggested merge at 'Cubic equation'. I've seen that you've proposed the merging of quartic function/equation and I think as long as we're consistent about article names, all will be fine.
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Weisstein, Eric W. "Cubic Formula." From MathWorld--A Wolfram Web Resource. that is, I hope, WP:Reliable sources and can be added at the External links of the article instead or above ˝Cubic Equation˝ of same author. Article contains 93 formula-lines referring to:
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I notice a conspicuous lack of citations throughout this page. If anyone knows where published forms of what is printed in this article can be found and is willing to go though and create citations, that would greatly improve the article as a whole.
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After seven Iterations we have, Approximate value of one root is to three decimal places is 3.000. Therefore , we get ( x – 3 ) a factor of x³ - 6x² + 11x – 6 = 0. Hence, by using synthetic Division method , we can further calculate the other roots.
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Are you DVdm an administrator? If so, I assumed you to take care of the significance and correctness from mathematical point of view rather than to supervise how it is signed in and/or to count how many edits one needs to improve his contribution.
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Yesterday I added a discussion comment as to an external reference in fact discussing the solution of quartics and not cubics and the confusing statement regarding D, and today I see that my comment is deleted and the text body remains unaltered.
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To be a sufficiently accurate source it needs to come from a peer-reviewable source like a scholarly journal. Published books are not bound to this requirement and thus should not be used as a primary source of information. --Fusionshrimp
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I suspect you either have defined your variable u as an integer, or you have used integer division. Anyway, always use a good old calculator before you try a programming language you don't thoroughly understand.
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When I merged from , I noticed each had a different set of formulae for finding the roots. I chose the one that looked simpler, but thought it best to preserve this one as well (just in case the simpler one is
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As far as I know the work so carelessly removed is original research of myself and I am confident in its correctness but disappointed that nobody recognized its advantages in comparison to the article:
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This is crazy. The use of the constants is so inconsistent that an average reader would not be able to follow the derivations near the bottom of the page. The discriminant case, for example, is wrong.
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The business of Omar Khayyámasnd the ruler and compass it should be more he conjectured or some word like that rather than that he discovered that the cubic could not be solved using ruler and compass.
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4284:, so Tusi may belong ahead of Viete. A case analysis was done, so the discriminant remark may not be far off. The comments about numerical methods are interesting, but could suggest misstatements.
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I would not delete it but rather label the passages as needing the citations checked; perhaps a dubious tag pointing here or comments on each questionable reference. Google books has one of the
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Therefore , according to Newton - Raphson’s method at x◦ = 6. we will have a calculation chart which shows the possible iterations using the equation given below. ( 1.4 ) x1 = x◦ - f(x◦)/f’(x◦)
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I've spent the past 2 days coding up both C++ and Java methods using several different variants based on this article and can't get any of them to give the correct answers in all test cases.
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this case the truth is that almost fifty years ago, being not aware of Dickson’s or Chebyshev’s even Tartaglia’s work, I passingly – in order to get a solution of integral type dx/P
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What was previously said was that Cardano broke his promise to Tartaglia. By modern standards, this is incorrect. The promise he made to Tartaglia was that he would not publish
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I haven't seen any discussion of the merge and so this may be a moot point that has already been demonstrated, but I would just like to point out the following inconsistency:
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Dickson, L. E. ˝A New solution of the Cubic Equation.˝ Amer. Math Monthly 5, 38.39, 1898 or Elementary Theory of Equations. New York: Wiley, pp. 36-37, 1914 as well as to
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we have a beautiful inequality , which can be further solved by using any iterative method. By using sign scheme we can extract the boundaries for the equation 1.1.
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case too. Tested and working, found on an Italian math paper. Sorry for my bad English and format. Please, make it more readable if you mind... Heavymachinegun
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2Q*q=\begin{vmatrix}0&3a&2b\\3a&b&c\\b&c&3d\end{vmatrix}= 9abc-27a^2d-2b^3, P*p=\begin{vmatrix}b&3a\\c&b\end{vmatrix}=b^2-3ac
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Four weeks passed by but I did not receive email as said – therefore I must replay here. Nevertheless, meanwhile I kept searching that resulted with:
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4Y = ±3X + T cuts Y = X^3. Shifting y-axis right for b/(3a) and sliding 4Y = ±3X up for T/4 at single graph can be presented all of 6 equation types:
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4y^3 + 3sgn(p) = (3/|p|)^1.5*q/2 = C (as a matter of fact a correction of Chebyshev one z^3 – 3z = q*p^1.5) along with roots for sgn(p) = +1 at (80):
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The first root given under the monic formula does not appear to be a solution to the monic equation. I have not tested the other two root formulas.
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Now we can mentally write the roots of the given cubic equation. Factorize constant term in the equation (d/a) ; into three factors say, p , q , r.
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which means journals, books reliable newspapers or suchlike. Get your contribution into a jouyrnal first before trying to stick it into wikipedia.
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We can use many methods to solve this equation (1.1) but according to Numerical Analysis method , we require two boundaries to obtain roots.
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where δ, α, β, γ are arbitrary constants. The matter which is to be observed is given below. When we split 2 then we get after rearranging ,
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just stating that there is a published book that contradicts what you are saying. So what is your source for the information you provided?--
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I saw no source for what was previously up, and what I'm saying has a source for it, so I believe I'm justified in correcting what was said.
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Add the mention that angle trisection is another old cubic equation problem, even it has been understood much later as such a cubic problem.
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What is the point of having a wikipedia "by the people, for the people" if some moderator deletes text without addressing the points made?
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stop using this article talk page as a notepad unless you have a specific proposal to modify or add something to the article, that is
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Now we will check the conditions given in 8 and 9 which set of p , q , r satisfies these conditions are the roots of the equation.
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I don’t care either E.W. Weisstein or main notation will be chosen - the point is that WP article should be reedited since it is
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is not welcome on Knowledge (XXG). It must be published and peer reviewed first. We are not qualified to peer review your work.
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Knowledge (XXG) is a 21st century equivalent of the dark ages scriptures transcribed by moderating monks. Free - but moderated.
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Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, pp. 90-91, 106-107, and 414-417, 1996.
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Being unable to know what Omar Khayyám and Sharaf al-Dīn al-Tūsī have really done on the subject, I suggest the following:
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I took the formula as written, substituted into the monic cubic function as written using Mathematica and did not get zero.
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Yes, good catch. I have modified the text to: "If b-3ac=0, then the cubic's inflection point is the only critical point."
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2116:= ( x + 3 )( 2x + 7 ) – 2 . so, y³ = y( 2y + 1 ) – 2 . or, y³ - y( 2y + 1 ) + 1 = 0. Or, ( y – 1 )( y + 1 )( y – 2 ) = 0
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Solving equations 4, 5 , 6 which are linear equations in four variables, we will get the roots of the equation 1, i.e.,
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coeffs a,b,c,d, to `always' represent the original coeffs at the top of the page. To do otherwise is really confusing.
4109:, from which the general equations are derived, should be in English at the very least on the English encyclopaedia. --
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And we see that ( 1,2,3 ) satisfies these conditions so these are the roots of the equation , x³ + 6x² + 11x + 6 = 0.
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Replacing the paragraphs concerning the two authors by a sentence asserting that they have results on this subject
1145:{\displaystyle r={\sqrt {\left({\frac {3ac-b^{2}}{9a^{2}}}\right)^{3}+q^{2}}}={\sqrt {\frac {\Delta }{108a^{4}}}}}
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for a reference, but couldn't find any reproducing this exact form. Does anyone have a good reference to a book?
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do the doublechecking? Why don't we just remove the contribs? If/when someone wants to have something restored,
2484:, but a few lines of text would be great. Apologies to all mathematicians recoiling in horror at the thought of
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is presented as a tool to approximate the root of an equation when it is only a tool to evaluate a polynomial.
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Due to inability of cube rooting of complex numbers the roots of equation type where sign(b^2 – 3ac) = +1 : -->
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Maple (12, 13) can't simplify to zero either. But use pen and paper and check it manually. It works perfectly.
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That's an old and archived RfC. The point is still valid though, and his contribs need to be doublechecked.
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The roots of original equation are x_k = /(3a) corresponding to (84): x_i = 2(|p|/3)^(1/2)*y_i – a_2/3.
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the mathematically oriented. Is there an easier-to-understand version of this article at all in it?
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Hello, Stap aka 77.238.*.*, I have removed your contributions from this talk page. Can you please:
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from the function rather than the other way around. I agree the inconsistency is not good though. —
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Hoping my stubbornness would be considered as benevolent but not reasonless. Mladen Stambuk, Stap
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After getting the boundaries for the equation 1.3 we can further proceed using Iterative Method.
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expression surd(-8.,3) does produce -2. I don't know the corresponding function in Mathematica.
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305:{\displaystyle u={\sqrt{{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}.\quad (4)}
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M must be negative or zero to have real outcome. Otherwise it will be some complex number.
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And for the record, the only one of those that doesn't have a "function" page is Quintic:
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667:{\displaystyle y^{3}={-B \over 2A}\pm {\sqrt {\left({B \over 2A}\right)^{2}-{C \over A}}}}
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That is your turn to do as you have to do! I’m not going to bother any more. Regards Stap
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1543:{\displaystyle x_{3}=-{\frac {1}{2}}(s+t)-{\frac {b}{3a}}-{\frac {\sqrt {3}}{2}}(s-t)i.}
1428:{\displaystyle x_{2}=-{\frac {1}{2}}(s+t)-{\frac {b}{3a}}+{\frac {\sqrt {3}}{2}}(s-t)i,}
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can provide the proper sourcing. Woulnd't that be the proper way to take care of this?
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Note :- The only thing which must be taken care of is ( ax + b) ≠ 0 and c,a ≠ 0 , : -->
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Can anyone provide a C++/Java implementation based on this page that actually works?
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Taking the sign scheme of this inequation, The solution for the inequality we have ,
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402:{\displaystyle u={\sqrt{-{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}}
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If you are on this page, I have already responded to your comment on my talk page.
3776:. Although these equations are perfectly correct, we can hardly call these sources
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In using these methods always remove the coefficient of x³. It should be always 1.
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sign your comments on talk pages with 4 tildes (~~~~) at the end of your messages?
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Pull a factor 2 out of the square root and divide numerator and denominator by 2.
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Hence , equations 3 and 1’ are identical so we can equate coefficients , we get,
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Precise that doubling the cube is the simplest and oldest studied cubic equation
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495:{\displaystyle u^{3}=-{q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}
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If you wish to start a new discussion or revive an old one, please do so on the
202:{\displaystyle u^{3}={q \over 2}\pm {\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}
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As from the equation 1 , we can say that , it can factorized as shown below,
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4x^3 + 18x^2 + 135x = –175.5; 1084 where x_0 = –1.5; 4 or in Radical form:
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4x^3 + 18x^2 – 81x = 148.5; 220; 364.5; 545 where x_0 = –1.5; 4; 4.5; 5 and
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There are one intermediate case (T = 1 = P) but two marginal ones (T = 0).
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In the "Root-finding formula" section the solution is split for DELTA : -->
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own words work that is in books is a reasonable sort of thing to put in.
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For example: 3x^3+10x^2+14x+27=0 and normalised to x^3+3.33x^2+4.66x+3=0
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are pretty much horrendously ugly and useless plethoras of information.
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The article states that the roots x can be found from x=-p/3u + u - a/3
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i.e, we get this three equations simply to solve these cubic equations,
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0 < |9abc – 2b – 27ad|(b – 3ac) < 2 along with both graphs where
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http://demonstrations.wolfram.com/ParametersForPlottingACubicPolynomial
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4X^3 – 3X = 0; 143/432; 1; 793/432 where X_0 = 0; 11/12; 1; 13/12 and
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Later on I'll add in a few more dates and specifics from my source.
917:{\displaystyle ax^{3}+bx^{2}+cx+d=a(x-x_{1})(x-x_{2})(x-x_{3})=0,\,}
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Overhere (and with this reduced precision) this would normalise to
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2133:(1.1) ax³ + bx² + cx + d = 0 ; Where a , b , c , d € R and a ≠ 0.
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http://hk.knowledge.yahoo.com/question/question?qid=7007111502216
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Looks ok to me. Can you specify in detail how you tested this?
1964:
This method is best illustrated using the example give below ,
2417:{\displaystyle x={\frac {-2b\pm {\sqrt {4b^{2}-12ac\ }}}{6a}}}
2236:
Source : Dr. Nilaish's Journal for maths., ISSN 0974 - 3022 ,
2110:
Proceeding in the same way, ( x + 3 )³ = x³ + 9x² + 27x + 27.
25:
2066:(15) ( x + p ) = 0. (16) ( x + p ) = 1. (17) ( x + p ) = -1.
2848:
use the same definition of the term within the same article?
2541:
One way to do this is has been shown by Thomas Mueller at "
2021:
Now complete ( x³ + Ax^2 ) in cube ,say, ( x + p )³ then,
1868:
Method is shown by the help of the article given below :-
1864:
Four methods for solution of the cubic equation by Nilaish
1014:{\displaystyle q={\frac {9abc-27a^{2}d-2b^{3}}{54a^{3}}}}
505:
I think the latter is consistent with the earlier stated
4055:
Does not every cubic have exactly one inflection point?
3526:
simplify to zero in Maple (and probably in Mathematica)
1910:(3) x³ + ( α + β + γ )x² + ( αβ + βγ + γα )x + αβγ = 0.
1879:
ax³ + bx² + cx + d = 0 ; where a, b , c , d € R , a ≠ 0.
4226:
4160:
4052:"if b2-3ac=0, then the cubic has one inflection point"
2344:
I think that formula for x is totally wrong. Should be
3816:
sign in with your username Stap before you make edits?
2038:
Now in practice R.H.S of the equation 13 turns into ,
3705:
3670:
3640:
3580:
3556:
3536:
3480:
3418:
3360:
3340:
3320:
2919:
2545:" perhaps we should add this to the reference list?
2453:
Indeed.. Wasn't paying attention on that.. Thanks! --
2350:
1829:
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1266:
1214:
1165:
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936:
786:
580:
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324:
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131:
3530:
you avoid the principal roots pitfall by expressing
1951:
x³ + ( α + β + γ )x² + ( αβ + βγ + γα )x + αβγ = 0.
1624:
Is there any reason why cubics should be different?
1156:
where Δ is the discriminant defined above. Now, let
726:
is later used as magnitude in polar form to compute
2705:However, this cubic equation has 3 non-zero roots.
3721:
3691:
3656:
3626:
3562:
3542:
3486:
3466:
3404:
3346:
3326:
3302:
2416:
1841:
1542:
1427:
1312:
1246:
1194:
1144:
1013:
916:
666:
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494:
401:
304:
201:
2072:And the solution of the equation 1 is given by ,
1960:(8) p + q + r = (b/a). (9) pq + qr + pr = (c/a).
1823:I've written the complete formula that treat the
3917:|9abc – 2b^3 – 27a^2d |/2| b^2 – 3ac |^1.5 : -->
3627:{\displaystyle x=-{\frac {p}{3u}}+u-{a \over 3}}
2659:gives u=0, which results in a division by zero?
2103:So, y³ − y = 0. Therefore, x = { -2 , -1 , 3 }.
2576:Since u is defined as satisfying the condition
1970:Sum of coefficients = S = 1 + 6 + 11 + 6 = 24.
567:{\displaystyle y^{6}+qy^{3}-{p^{3} \over 27}=0}
315:But according to the equation some lines lower
3953:4X^3 + 3X = 0; 2519/432 where X_0 = 0; 11/12
3354:to be the second cube root. I then evaluated
2204:Using my analysis ; x³ - 6x² = -( 11x – 6 ) .
2130:Standard form of cubic equation is given by ,
8:
757:cubed plus a square which must be positive.
714:^3). This does not check the condition that
3570:and the coefficients (compare with section
2075:(18) x = { -p , ( -p + 1 ) , ( -p – 1 ) }.
2017:Now let , (11) (x³ + Ax^2 ) = − ( Bx + C )
2210:As per using my equation 1.2 we have now,
2035:(13) ( x + p )³ = − ( Bx + C ) + A’x + p³
1313:{\displaystyle x_{1}=s+t-{\frac {b}{3a}},}
4203:has been filed concerning the conduct of
3709:
3704:
3674:
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2349:
2198:Solve for x in , x³ - 6x² + 11x – 6 = 0.
1828:
1507:
1489:
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1449:
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136:
130:
2772:Now go and moderate my free expression.
2757:
574:when applying the quadratic ABC-formula
4223:misused sources here over several years
4022:almost worthless without explicit roots
3936:-determinant facilitating remembrance:
3884:Anything in wikipedia must be based on
3809:have a close look at Knowledge (XXG)'s
2195:One Example to illustrate my method :-
2093:But , ( x + 2 )³ = x³ + 6x² + 12x + 8.
912:
718:might be positive resulting in complex
3923:y = sinh and for sgn(p) = –1 at (83):
2142:(1.1) ax³ + bx² + cx + d = 0 ; a ≠ 0.
2024:(12) ( x + p )³ = x³ + Ax² + A’x + p³
44:Do not edit the contents of this page.
3760:Proper source for Standard and Monic?
3657:{\displaystyle -{\frac {\alpha }{3}}}
3634:, and where u plays the role of your
2122:Note : 2( x + 3 ) – 6 + 7 = (2y + 1)
2006:A = (b/a) ; B = (c/a) and C = (d/a).
1904:(2) δ( x + α )( x + β )( x + γ ) = 0
7:
3722:{\displaystyle -{\frac {\beta }{3}}}
2843:Inconsistent Discriminant Definition
2758:Why I don't have a wikipedia account
2515:Although I should say that this and
2176:( x + d/c )( x + b/a ) ≤ 0. c,a ≠ 0
2173:Iff , ( ax + b) ≠ 0 . Now we have ,
749:If the determinant is negative then
734:. Complex magnitude is nonsensical.
4300:User talk:Tobby72#Misuse of sources
3764:The current source for the section
1973:Constant term in the equation = 6.
1819:The (complete) root-finding formula
2082:A few examples for illustration :-
2027:where , Ax² = 3px² and A’x = 3p²x
1830:
1121:
24:
3768:(and hence the monic formula) is
3405:{\displaystyle x^{3}+ax^{2}+bx+c}
2266:Lagrange solvents, my oh, my....
2182:(1.3) ( x + d/c )( x + b/a ) ≤ 0
2031:Now put 7.11 in 7.12 and we get,
1874:We have general cubic equation ,
118:Inconsistency in Cardano's method
3828:no original research by yourself
3692:{\displaystyle -{\frac {p}{3u}}}
2594:, which reduces equation (2) to
2424:.. Correct me, if I'm wrong.. --
1976:Now we have , 6 = 1.2.3 = 1.1.6
753:is negative. The determinant is
29:
3572:Cubic function#Cardano's method
2238:http://nilaish.livejournal.com/
2170:Or , ( cx + d )( ax + b ) ≤ 0.
1943:x³ + (b/a)x² + (c/a)x + d/a = 0
1886:x³ + (b/a)x² + (c/a)x + d/a = 0
1247:{\displaystyle t={\sqrt{q-r}}.}
3516:Right. Mathematica... Maple...
3334:to be the first cube root and
2480:does this graphically through
2213:Or , ( 11x – 6 )( x – 6 ) ≤ 0
2154:Or , x = √(-M). , where M = .
2148:Or, ax³ + bx² = - ( cx + d ).
2139:We have from equation (1.1) ,
1967:F(x) = x³ + 6x² + 11x + 6 = 0
1939:As from equation 1’ we have ,
1531:
1519:
1483:
1471:
1416:
1404:
1368:
1356:
1195:{\displaystyle s={\sqrt{q+r}}}
900:
881:
878:
859:
856:
837:
299:
293:
1:
4189:03:34, 26 February 2010 (UTC)
4173:08:45, 25 February 2010 (UTC)
4139:06:57, 22 February 2010 (UTC)
4096:23:12, 21 December 2009 (UTC)
4079:21:44, 21 December 2009 (UTC)
4040:07:32, 10 December 2009 (UTC)
3898:12:03, 10 November 2009 (UTC)
3875:08:49, 10 November 2009 (UTC)
3799:Attention Step aka 77.238.*.*
2863:Monic formula of roots error?
2463:19:45, 31 December 2008 (UTC)
2449:10:16, 30 December 2008 (UTC)
2434:22:39, 29 December 2008 (UTC)
2332:01:31, 18 November 2008 (UTC)
2107:Ex.2. x³ + 7x² + 14x + 8 = 0
2087:Ex1. x³ + 6x² + 11x + 6 = 0.
2041:(14) ( x + p )³ = ( x + p ).
1947:As from equation 3 we have ,
1792:17:15, 28 December 2008 (UTC)
291:
4298:Yes, possibly, but see also
4005:07:59, 7 December 2009 (UTC)
3986:00:42, 7 December 2009 (UTC)
3967:23:43, 6 December 2009 (UTC)
3846:07:09, 21 October 2009 (UTC)
2911:Specifically, I considered:
2834:20:36, 3 February 2009 (UTC)
2812:19:10, 22 January 2009 (UTC)
2752:17:29, 22 January 2009 (UTC)
2700:17:29, 22 January 2009 (UTC)
2654:17:29, 22 January 2009 (UTC)
2610:17:29, 22 January 2009 (UTC)
2013:(10) x³ + Ax² + Bx + C = 0.
1842:{\displaystyle \Delta <0}
1741:20:48, 6 December 2007 (UTC)
1712:20:44, 6 December 2007 (UTC)
1664:20:30, 6 December 2007 (UTC)
1642:20:19, 6 December 2007 (UTC)
767:19:30, 6 December 2009 (UTC)
744:18:51, 6 December 2009 (UTC)
113:20:08, 6 December 2007 (UTC)
4119:07:36, 5 January 2010 (UTC)
3849:= Stap answers to DVdm ==
3794:13:47, 10 August 2009 (UTC)
3750:21:17, 10 August 2009 (UTC)
3504:19:39, 10 August 2009 (UTC)
2675:x + 3.33 x + 4.66 x + 3 = 0
2493:18:50, 8 January 2009 (UTC)
2306:15:09, 22 August 2008 (UTC)
2096:= - ( 11x + 6 ) + 12x + 8.
2090:ð x³ + 6x² = - ( 11x + 6 )
2003:From the equation 1’ let ,
1554:Correctness of the formulas
97:or something!) See below: —
4417:
4219:198 in the number of edits
2894:12:34, 7 August 2009 (UTC)
2879:22:10, 5 August 2009 (UTC)
2858:17:45, 13 April 2009 (UTC)
2250:05:18, 4 August 2008 (UTC)
2119:So, x = { -2 , 4 , ± 1 }.
2045:So, we put ( x + p ) = y.
1989:And ( 1,1,6) is rejected.
698:another inconsistency here
677:another inconsistency here
4401:09:43, 11 June 2010 (UTC)
4384:09:34, 11 June 2010 (UTC)
4367:03:35, 11 June 2010 (UTC)
4312:22:06, 10 June 2010 (UTC)
4294:21:56, 10 June 2010 (UTC)
4276:21:02, 10 June 2010 (UTC)
4254:20:55, 10 June 2010 (UTC)
4239:19:14, 10 June 2010 (UTC)
2684:. Overhere it results in
2535:20:16, 2 March 2009 (UTC)
2509:20:11, 2 March 2009 (UTC)
2048:So the equation becomes,
1871:Abstract & Article :
1811:20:02, 2 March 2009 (UTC)
1776:03:20, 4 March 2008 (UTC)
1751:Cardano/Tartaglia Contest
692:18:36, 30 July 2008 (UTC)
4374:looks reasonable to me.
3929:y = cos for |C| <1.
3766:General formula of roots
2555:11:31, 1 July 2009 (UTC)
1859:23:49, 4 July 2008 (UTC)
1572:15:52, 19 May 2010 (UTC)
4024:for equation type where
3563:{\displaystyle \alpha }
3327:{\displaystyle \alpha }
2802:Consider it done :-) -
2471:How to use these things
2285:17:46, 2 May 2009 (UTC)
2222:Now testing f(a) =: -->
2145:And a , b , c , d € R.
1931:(7) x = {-α, -β, -γ }.
4227:September/October 2008
3723:
3693:
3658:
3628:
3564:
3544:
3543:{\displaystyle \beta }
3488:
3468:
3406:
3348:
3347:{\displaystyle \beta }
3328:
3304:
2418:
1843:
1544:
1429:
1314:
1248:
1196:
1146:
1015:
918:
668:
568:
496:
403:
306:
203:
117:
4258:Question: why should
4163:on your talk page. -
3927:y = –cosh for C ≤ –1
3724:
3694:
3659:
3629:
3565:
3545:
3489:
3469:
3407:
3349:
3329:
3305:
2419:
1844:
1545:
1430:
1315:
1249:
1197:
1147:
1016:
919:
669:
569:
497:
404:
307:
204:
42:of past discussions.
4201:request for comments
4028:9abc – 2b – 27ad = 0
3932:determinant and its
3860:WP:Original research
3811:talk page guidelines
3703:
3668:
3638:
3578:
3554:
3534:
3478:
3416:
3358:
3338:
3318:
2917:
2348:
2223:f ( 6/11 ) <: -->
1827:
1442:
1327:
1264:
1212:
1163:
1031:
934:
784:
578:
509:
419:
322:
214:
129:
88:Root-finding Formula
3972:As was said before
3925:y = cosh for C ≥ 1
3886:WP:Reliable sources
2678:does not result in
2478:quadratic equations
1257:The solutions are
18:Talk:Cubic function
3719:
3689:
3654:
3624:
3560:
3540:
3484:
3464:
3402:
3344:
3324:
3300:
3298:
2560:question about u=0
2414:
2291:Correct equal case
2240:, Vol.1, No. 8,7.
2113:= 2x² + 13x + 19.
1839:
1679:Quadratic function
1540:
1425:
1310:
1244:
1192:
1142:
1011:
914:
913:
702:When discriminant
664:
564:
492:
399:
302:
292:
199:
4331:doubling the cube
4326:doubling the cube
4195:Misuse of sources
4146:comment added by
4082:
4065:comment added by
3974:Original research
3881:Original research
3772:and it points to
3717:
3687:
3652:
3622:
3603:
3522:With numerics it
3487:{\displaystyle x}
3462:
3449:
3436:
3294:
3288:
3282:
3144:
3123:
3117:
3111:
2973:
2952:
2824:comment added by
2793:
2779:comment added by
2732:
2718:comment added by
2525:comment added by
2412:
2401:
2400:
2322:comment added by
2275:comment added by
1924:(6) αβγ = (d/a).
1920:(5) ∑αβ = (c/a).
1738:
1710:
1662:
1639:
1517:
1513:
1502:
1469:
1402:
1398:
1387:
1354:
1305:
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197:
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153:
111:
85:
84:
54:
53:
48:current talk page
4408:
4335:angle trisection
4155:
4081:
4059:
4048:Inflection Point
3824:properly sourced
3728:
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3725:
3720:
3718:
3710:
3698:
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2792:
2773:
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2712:
2627:x + 3.33 x + 4.6
2537:
2517:related articles
2488:maths.... :-) --
2423:
2421:
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2415:
2413:
2411:
2403:
2402:
2399:
2386:
2385:
2373:
2358:
2334:
2287:
2058:y²( y − 1 ) = 0.
1917:(4) ∑α = (b/a).
1848:
1846:
1845:
1840:
1737:
1732:
1729:
1700:
1689:Quintic function
1684:Quartic function
1652:
1638:
1633:
1630:
1561:quartic function
1549:
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140:
122:It is said that
101:
63:
56:
55:
33:
32:
26:
4416:
4415:
4411:
4410:
4409:
4407:
4406:
4405:
4197:
4141:
4126:
4104:
4060:
4050:
4016:
3801:
3762:
3701:
3700:
3679:
3666:
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3595:
3576:
3575:
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3531:
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3413:
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3316:
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3194:
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3189:
3153:
3149:
3129:
3126:
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3027:
3023:
3019:
3018:
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2958:
2955:
2954:
2937:
2924:
2915:
2914:
2865:
2845:
2819:
2774:
2760:
2713:
2638:, resulting in
2582:, your what-if
2567:What if u==0??
2562:
2520:
2473:
2404:
2377:
2359:
2346:
2345:
2342:
2317:
2313:
2293:
2270:
2260:
1866:
1825:
1824:
1821:
1753:
1733:
1725:
1698:Celtic Minstrel
1674:Linear function
1650:Celtic Minstrel
1634:
1626:
1579:
1556:
1494:
1445:
1440:
1439:
1379:
1330:
1325:
1324:
1297:
1267:
1262:
1261:
1223:
1210:
1209:
1174:
1161:
1160:
1128:
1124:
1103:
1077:
1073:
1062:
1049:
1043:
1042:
1029:
1028:
998:
994:
983:
964:
945:
932:
931:
890:
868:
846:
806:
790:
782:
781:
775:
700:
679:
629:
620:
619:
604:
596:
581:
576:
575:
543:
528:
512:
507:
506:
475:
455:
422:
417:
416:
374:
354:
333:
320:
319:
263:
243:
225:
212:
211:
182:
162:
132:
127:
126:
120:
99:Celtic Minstrel
90:
59:
30:
22:
21:
20:
12:
11:
5:
4414:
4412:
4404:
4403:
4387:
4386:
4370:
4369:
4355:
4352:
4321:
4320:
4319:
4318:
4317:
4316:
4315:
4314:
4196:
4193:
4192:
4191:
4176:
4175:
4125:
4122:
4103:
4100:
4099:
4098:
4049:
4046:
4044:
4032:77.238.206.113
4029:
4027:
4025:
4014:
4010:
4009:
4008:
4007:
3989:
3988:
3956:
3954:
3952:
3950:
3948:
3946:
3944:
3941:
3939:
3937:
3930:
3928:
3926:
3924:
3922:
3920:
3901:
3900:
3867:77.238.214.167
3862:and email....
3853:
3850:
3848:
3834:
3833:
3832:
3831:
3820:
3817:
3814:
3800:
3797:
3780:. I have been
3761:
3758:
3757:
3756:
3755:
3754:
3753:
3752:
3738:
3734:
3716:
3713:
3708:
3685:
3682:
3678:
3673:
3651:
3648:
3643:
3621:
3618:
3613:
3610:
3607:
3601:
3598:
3594:
3589:
3586:
3583:
3559:
3539:
3520:
3517:
3509:
3508:
3507:
3506:
3496:128.135.226.20
3483:
3461:
3458:
3453:
3448:
3445:
3440:
3435:
3432:
3427:
3424:
3421:
3401:
3398:
3395:
3392:
3389:
3384:
3380:
3376:
3373:
3368:
3364:
3343:
3323:
3314:and I defined
3312:
3311:
3310:
3292:
3287:
3279:
3274:
3270:
3267:
3264:
3259:
3255:
3250:
3245:
3242:
3237:
3232:
3228:
3225:
3222:
3219:
3216:
3213:
3210:
3205:
3201:
3197:
3193:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3160:
3156:
3152:
3143:
3140:
3135:
3132:
3130:
3128:
3127:
3121:
3116:
3108:
3103:
3099:
3096:
3093:
3088:
3084:
3079:
3074:
3071:
3066:
3061:
3057:
3054:
3051:
3048:
3045:
3042:
3039:
3034:
3030:
3026:
3022:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2989:
2985:
2981:
2972:
2969:
2964:
2961:
2959:
2957:
2956:
2951:
2948:
2943:
2940:
2938:
2936:
2931:
2927:
2923:
2922:
2906:
2905:
2904:
2903:
2897:
2896:
2871:128.135.226.20
2864:
2861:
2844:
2841:
2839:
2815:
2814:
2759:
2756:
2755:
2754:
2703:
2702:
2657:
2656:
2613:
2612:
2561:
2558:
2539:
2538:
2527:128.194.39.250
2512:
2511:
2501:128.194.39.250
2482:a clever image
2472:
2469:
2468:
2467:
2466:
2465:
2410:
2407:
2398:
2395:
2392:
2389:
2384:
2380:
2376:
2371:
2368:
2365:
2362:
2356:
2353:
2341:
2336:
2324:75.157.213.174
2312:
2309:
2292:
2289:
2269:Gaddy Alcalá
2259:
2256:
2254:
2235:
2230:
2164:Which means ,
2157:
2106:
2102:
2086:
2078:
2069:
2062:
2051:
2044:
2034:
2030:
2020:
2016:
2010:Hence we get,
2009:
1986:
1982:
1963:
1950:
1927:
1923:
1913:
1903:
1865:
1862:
1838:
1835:
1832:
1820:
1817:
1816:
1815:
1814:
1813:
1803:128.194.39.250
1795:
1794:
1752:
1749:
1748:
1747:
1746:
1745:
1744:
1743:
1715:
1714:
1694:
1693:
1692:
1691:is a redirect.
1686:
1681:
1676:
1667:
1666:
1578:
1575:
1555:
1552:
1551:
1550:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1516:
1512:
1506:
1500:
1497:
1493:
1488:
1485:
1482:
1479:
1476:
1473:
1468:
1465:
1460:
1457:
1452:
1448:
1436:
1435:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1401:
1397:
1391:
1385:
1382:
1378:
1373:
1370:
1367:
1364:
1361:
1358:
1353:
1350:
1345:
1342:
1337:
1333:
1321:
1320:
1309:
1303:
1300:
1296:
1291:
1288:
1285:
1282:
1279:
1274:
1270:
1255:
1254:
1243:
1237:
1232:
1229:
1226:
1220:
1217:
1203:
1202:
1188:
1183:
1180:
1177:
1171:
1168:
1155:
1153:
1152:
1135:
1131:
1127:
1123:
1117:
1110:
1106:
1102:
1097:
1092:
1084:
1080:
1076:
1069:
1065:
1061:
1058:
1055:
1052:
1046:
1039:
1036:
1022:
1021:
1005:
1001:
997:
990:
986:
982:
979:
976:
971:
967:
963:
960:
957:
954:
951:
948:
942:
939:
925:
924:
911:
908:
905:
902:
897:
893:
889:
886:
883:
880:
875:
871:
867:
864:
861:
858:
853:
849:
845:
842:
839:
836:
833:
830:
827:
824:
821:
818:
813:
809:
805:
802:
797:
793:
789:
774:
771:
770:
769:
699:
696:
694:
678:
675:
659:
656:
651:
646:
641:
635:
632:
628:
623:
616:
610:
607:
602:
599:
593:
588:
584:
563:
560:
555:
550:
546:
540:
535:
531:
527:
524:
519:
515:
503:
502:
487:
482:
478:
472:
467:
462:
458:
450:
445:
442:
437:
434:
429:
425:
410:
409:
395:
386:
381:
377:
371:
366:
361:
357:
349:
344:
341:
336:
330:
327:
313:
312:
301:
298:
295:
290:
284:
275:
270:
266:
260:
255:
250:
246:
238:
233:
230:
222:
219:
209:
194:
189:
185:
179:
174:
169:
165:
157:
152:
149:
144:
139:
135:
119:
116:
89:
86:
83:
82:
77:
74:
69:
64:
52:
51:
34:
23:
15:
14:
13:
10:
9:
6:
4:
3:
2:
4413:
4402:
4398:
4394:
4389:
4388:
4385:
4381:
4377:
4372:
4371:
4368:
4364:
4360:
4356:
4353:
4350:
4349:
4348:
4345:
4343:
4342:Horner scheme
4338:
4336:
4332:
4327:
4313:
4309:
4305:
4301:
4297:
4296:
4295:
4291:
4287:
4283:
4279:
4278:
4277:
4273:
4269:
4265:
4261:
4257:
4256:
4255:
4251:
4247:
4243:
4242:
4241:
4240:
4236:
4232:
4228:
4224:
4220:
4216:
4213:
4210:
4206:
4202:
4194:
4190:
4186:
4182:
4178:
4177:
4174:
4170:
4166:
4162:
4158:
4157:
4156:
4153:
4149:
4148:71.118.39.165
4145:
4140:
4136:
4132:
4123:
4121:
4120:
4116:
4112:
4111:121.44.147.29
4108:
4101:
4097:
4093:
4089:
4085:
4084:
4083:
4080:
4076:
4072:
4068:
4064:
4056:
4053:
4047:
4045:
4042:
4041:
4037:
4033:
4023:
4018:
4006:
4002:
3998:
3993:
3992:
3991:
3990:
3987:
3983:
3979:
3975:
3971:
3970:
3969:
3968:
3964:
3960:
3959:77.238.216.76
3935:
3914:
3911:
3908:
3904:
3899:
3895:
3891:
3887:
3882:
3879:
3878:
3877:
3876:
3872:
3868:
3863:
3861:
3856:
3847:
3843:
3839:
3836:Thank you. -
3829:
3825:
3821:
3818:
3815:
3812:
3808:
3807:
3806:
3805:
3804:
3798:
3796:
3795:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3759:
3751:
3747:
3743:
3739:
3735:
3732:
3714:
3711:
3706:
3683:
3680:
3676:
3671:
3649:
3646:
3641:
3619:
3616:
3611:
3608:
3605:
3599:
3596:
3592:
3587:
3584:
3581:
3573:
3557:
3537:
3529:
3525:
3521:
3518:
3515:
3514:
3513:
3512:
3511:
3510:
3505:
3501:
3497:
3481:
3459:
3456:
3451:
3446:
3443:
3438:
3433:
3430:
3425:
3422:
3419:
3399:
3396:
3393:
3390:
3387:
3382:
3378:
3374:
3371:
3366:
3362:
3341:
3321:
3313:
3290:
3285:
3277:
3272:
3268:
3265:
3262:
3257:
3253:
3248:
3243:
3240:
3235:
3230:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3203:
3199:
3195:
3191:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3158:
3154:
3150:
3141:
3138:
3133:
3131:
3119:
3114:
3106:
3101:
3097:
3094:
3091:
3086:
3082:
3077:
3072:
3069:
3064:
3059:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3032:
3028:
3024:
3020:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2987:
2983:
2979:
2970:
2967:
2962:
2960:
2949:
2946:
2941:
2939:
2934:
2929:
2925:
2913:
2912:
2910:
2909:
2908:
2907:
2901:
2900:
2899:
2898:
2895:
2891:
2887:
2883:
2882:
2881:
2880:
2876:
2872:
2868:
2862:
2860:
2859:
2855:
2851:
2850:John Lawrence
2842:
2840:
2837:
2835:
2831:
2827:
2826:76.75.119.183
2823:
2813:
2809:
2805:
2801:
2800:
2796:
2795:
2794:
2790:
2786:
2782:
2781:212.20.240.70
2778:
2770:
2767:
2764:
2753:
2749:
2745:
2740:
2739:
2735:
2734:
2733:
2729:
2725:
2721:
2720:212.20.240.70
2717:
2709:
2706:
2701:
2697:
2693:
2689:
2688:
2683:
2682:
2677:
2676:
2671:
2667:
2666:
2662:
2661:
2660:
2655:
2651:
2647:
2643:
2642:
2637:
2636:
2634:
2630:
2623:
2622:
2618:
2617:
2616:
2611:
2607:
2603:
2599:
2598:
2593:
2592:
2588:demands that
2587:
2586:
2581:
2580:
2575:
2574:
2570:
2569:
2568:
2565:
2559:
2557:
2556:
2552:
2548:
2544:
2536:
2532:
2528:
2524:
2518:
2514:
2513:
2510:
2506:
2502:
2497:
2496:
2495:
2494:
2491:
2487:
2483:
2479:
2470:
2464:
2460:
2456:
2452:
2451:
2450:
2446:
2442:
2438:
2437:
2436:
2435:
2431:
2427:
2408:
2405:
2396:
2393:
2390:
2387:
2382:
2378:
2374:
2369:
2366:
2363:
2360:
2354:
2351:
2340:
2337:
2335:
2333:
2329:
2325:
2321:
2310:
2308:
2307:
2303:
2299:
2290:
2288:
2286:
2282:
2278:
2277:200.90.51.187
2274:
2267:
2264:
2257:
2255:
2252:
2251:
2247:
2243:
2239:
2233:
2228:
2225:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2127:
2123:
2120:
2117:
2114:
2111:
2108:
2104:
2100:
2099:= ( x + 2 ).
2097:
2094:
2091:
2088:
2084:
2083:
2079:
2076:
2073:
2070:
2067:
2064:
2060:
2059:
2055:
2049:
2046:
2042:
2039:
2036:
2032:
2028:
2025:
2022:
2018:
2014:
2011:
2007:
2004:
2001:
2000:
1996:
1994:
1990:
1987:
1984:
1980:
1977:
1974:
1971:
1968:
1965:
1961:
1958:
1955:
1952:
1948:
1945:
1944:
1940:
1937:
1936:
1932:
1929:
1925:
1921:
1918:
1915:
1911:
1908:
1905:
1901:
1898:
1897:
1893:
1892:
1888:
1887:
1882:
1880:
1875:
1872:
1869:
1863:
1861:
1860:
1856:
1852:
1851:87.10.134.182
1836:
1833:
1818:
1812:
1808:
1804:
1799:
1798:
1797:
1796:
1793:
1789:
1785:
1780:
1779:
1778:
1777:
1773:
1769:
1764:
1761:
1756:
1750:
1742:
1739:
1736:
1730:
1728:
1721:
1720:
1719:
1718:
1717:
1716:
1713:
1708:
1704:
1699:
1695:
1690:
1687:
1685:
1682:
1680:
1677:
1675:
1672:
1671:
1669:
1668:
1665:
1660:
1656:
1651:
1646:
1645:
1644:
1643:
1640:
1637:
1631:
1629:
1622:
1621:
1620:
1614:
1613:
1612:
1606:
1605:
1604:
1598:
1597:
1596:
1590:
1589:
1588:
1582:
1576:
1574:
1573:
1569:
1565:
1562:
1553:
1537:
1534:
1528:
1525:
1522:
1514:
1510:
1504:
1498:
1495:
1491:
1486:
1480:
1477:
1474:
1466:
1463:
1458:
1455:
1450:
1446:
1438:
1437:
1422:
1419:
1413:
1410:
1407:
1399:
1395:
1389:
1383:
1380:
1376:
1371:
1365:
1362:
1359:
1351:
1348:
1343:
1340:
1335:
1331:
1323:
1322:
1307:
1301:
1298:
1294:
1289:
1286:
1283:
1280:
1277:
1272:
1268:
1260:
1259:
1258:
1241:
1235:
1230:
1227:
1224:
1218:
1215:
1208:
1207:
1206:
1186:
1181:
1178:
1175:
1169:
1166:
1159:
1158:
1157:
1133:
1129:
1125:
1115:
1108:
1104:
1100:
1095:
1090:
1082:
1078:
1074:
1067:
1063:
1059:
1056:
1053:
1050:
1044:
1037:
1034:
1027:
1026:
1025:
1003:
999:
995:
988:
984:
980:
977:
974:
969:
965:
961:
958:
955:
952:
949:
946:
940:
937:
930:
929:
928:
909:
906:
903:
895:
891:
887:
884:
873:
869:
865:
862:
851:
847:
843:
840:
834:
831:
828:
825:
822:
819:
816:
811:
807:
803:
800:
795:
791:
787:
780:
779:
778:
772:
768:
764:
760:
756:
752:
748:
747:
746:
745:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
697:
695:
693:
689:
685:
676:
674:
657:
654:
649:
644:
639:
633:
630:
626:
621:
614:
608:
605:
600:
597:
591:
586:
582:
561:
558:
553:
548:
544:
538:
533:
529:
525:
522:
517:
513:
485:
480:
476:
470:
465:
460:
456:
448:
443:
440:
435:
432:
427:
423:
415:
414:
413:
393:
384:
379:
375:
369:
364:
359:
355:
347:
342:
339:
334:
328:
325:
318:
317:
316:
296:
288:
282:
273:
268:
264:
258:
253:
248:
244:
236:
231:
228:
220:
217:
210:
192:
187:
183:
177:
172:
167:
163:
155:
150:
147:
142:
137:
133:
125:
124:
123:
115:
114:
109:
105:
100:
96:
87:
81:
78:
75:
73:
70:
68:
65:
62:
58:
57:
49:
45:
41:
40:
35:
28:
27:
19:
4346:
4339:
4322:
4263:
4259:
4211:
4198:
4181:71.108.11.25
4161:this comment
4131:71.108.26.48
4127:
4124:Introduction
4105:
4057:
4054:
4051:
4043:
4021:
4019:
4011:
3933:
3915:
3912:
3909:
3905:
3902:
3864:
3858:....Removed
3857:
3835:
3826:and that is
3802:
3777:
3763:
3730:
3550:in terms of
3527:
3523:
2869:
2866:
2846:
2838:
2816:
2798:
2797:
2771:
2768:
2765:
2761:
2737:
2736:
2710:
2707:
2704:
2686:
2685:
2680:
2679:
2674:
2673:
2669:
2664:
2663:
2658:
2640:
2639:
2632:
2628:
2626:
2625:
2620:
2619:
2614:
2596:
2595:
2590:
2589:
2584:
2583:
2578:
2577:
2572:
2571:
2566:
2563:
2540:
2485:
2474:
2343:
2314:
2294:
2268:
2265:
2261:
2253:
2234:
2229:
2226:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2200:
2197:
2194:
2191:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2161:(1.2) M ≤ 0
2160:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2125:
2124:
2121:
2118:
2115:
2112:
2109:
2105:
2101:
2098:
2095:
2092:
2089:
2085:
2081:
2080:
2077:
2074:
2071:
2068:
2065:
2061:
2057:
2053:
2050:
2047:
2043:
2040:
2037:
2033:
2029:
2026:
2023:
2019:
2015:
2012:
2008:
2005:
2002:
1998:
1997:
1992:
1991:
1988:
1985:
1981:
1978:
1975:
1972:
1969:
1966:
1962:
1959:
1956:
1953:
1949:
1946:
1942:
1941:
1938:
1934:
1933:
1930:
1926:
1922:
1919:
1916:
1912:
1909:
1906:
1902:
1899:
1895:
1894:
1890:
1889:
1885:
1883:
1878:
1876:
1873:
1870:
1867:
1822:
1765:
1759:
1757:
1754:
1734:
1726:
1635:
1627:
1623:
1618:
1615:
1610:
1607:
1602:
1599:
1594:
1591:
1586:
1583:
1580:
1557:
1256:
1204:
1154:
1023:
926:
776:
754:
750:
731:
727:
723:
719:
715:
711:
707:
703:
701:
680:
504:
411:
314:
121:
94:
91:
60:
43:
37:
4142:—Preceding
4102:Source Nine
4061:—Preceding
3733:root terms.
2820:—Preceding
2775:—Preceding
2714:—Preceding
2579:3uv + p = 0
2521:—Preceding
2490:Matt's talk
2318:—Preceding
2271:—Preceding
2258:Observation
2207:So, x = √.
2201:Solution :
2151:So, x = √.
1957:Such that,
777:If we have
773:The formula
36:This is an
4340:Similarly
4282:references
2339:Derivative
2219:So, x € .
2126:Method 4 :
2054:y³ − y = 0
1891:Solution :
1760:Tartaglia'
1593:Quadratic
4205:Jagged 85
4058:~JacekW
2687:u = 0.377
2641:u = 0.173
2597:t + q = 0
2547:Halothane
2311:Citations
1727:asyndeton
1628:asyndeton
704:Δ <= 0
684:Halothane
80:Archive 5
72:Archive 3
67:Archive 2
61:Archive 1
4359:D.Lazard
4215:contribs
4144:unsigned
4075:contribs
4063:unsigned
3699:of your
3574:, where
3528:provided
2822:unsigned
2789:contribs
2777:unsigned
2728:contribs
2716:unsigned
2672:version
2523:unsigned
2320:unsigned
2273:unsigned
1999:Method 3
1935:Method 2
1896:Method 1
1768:Fephisto
1707:contribs
1659:contribs
1619:equation
1617:Qunitic
1611:equation
1609:Quartic
1603:function
1595:equation
1587:equation
1564:D.Lazard
736:Jhshukla
710:= sqrt(-
108:contribs
4329:namely
4246:Tobby72
4231:Tobby72
3782:looking
3740:Enjoy,
2799:==: -->
2738:==: -->
2665:==: -->
2621:==: -->
2573:==: -->
2242:nilaish
1993:Note :-
1881:Also ,
1585:Linear
39:archive
4067:JacekW
3778:proper
3664:, and
2179:i.e.,
1784:Terets
1601:Cubic
730:&
3919:(78):
2681:u = 0
2668:Even
2591:p = 0
2585:u = 0
2486:using
2298:Wentu
2167:≤ 0.
1884:(1’)
1577:Merge
95:wrong
16:<
4397:talk
4393:Dmcq
4380:talk
4376:Dmcq
4363:talk
4333:and
4308:talk
4304:DVdm
4290:talk
4286:Glrx
4272:talk
4268:DVdm
4264:they
4250:talk
4235:talk
4209:talk
4185:talk
4169:talk
4165:DVdm
4159:See
4152:talk
4135:talk
4115:talk
4092:talk
4088:DVdm
4071:talk
4036:talk
4001:talk
3997:Dmcq
3982:talk
3978:Dmcq
3963:talk
3894:talk
3890:Dmcq
3871:talk
3865:Stap
3842:talk
3838:DVdm
3790:talk
3786:DVdm
3774:this
3770:this
3746:talk
3742:DVdm
3731:both
3524:does
3500:talk
3412:for
2890:talk
2886:DVdm
2875:talk
2854:talk
2830:talk
2808:talk
2804:DVdm
2785:talk
2748:talk
2744:DVdm
2724:talk
2696:talk
2692:DVdm
2670:your
2650:talk
2646:DVdm
2631:x +
2606:talk
2602:DVdm
2551:talk
2531:talk
2505:talk
2459:talk
2445:talk
2441:DVdm
2430:talk
2328:talk
2302:talk
2281:talk
2246:talk
2056:Or,
2052:Or ,
1877:(1)
1855:talk
1834:<
1807:talk
1788:talk
1772:talk
1735:talk
1703:talk
1655:talk
1636:talk
1568:talk
1205:and
1024:and
927:let
763:talk
759:Dmcq
740:talk
688:talk
104:talk
3934:sub
2635:= 0
2455:BiH
2426:BiH
2224:0.
2189:0.
1126:108
412:so
4399:)
4382:)
4365:)
4337:.
4310:)
4302:-
4292:)
4274:)
4260:we
4252:)
4237:)
4229:.
4199:A
4187:)
4171:)
4154:)
4137:)
4117:)
4094:)
4077:)
4073:•
4038:)
4003:)
3984:)
3965:)
3896:)
3873:)
3844:)
3792:)
3748:)
3712:β
3707:−
3672:−
3647:α
3642:−
3612:−
3588:−
3558:α
3538:β
3502:)
3494:.
3457:β
3452:−
3444:α
3439:−
3426:−
3342:β
3322:α
3263:−
3241:−
3224:27
3209:−
3185:−
3179:27
3164:−
3134:−
3092:−
3070:−
3053:27
3038:−
3008:27
2993:−
2963:−
2942:−
2892:)
2877:)
2856:)
2832:)
2810:)
2791:)
2787:•
2750:)
2730:)
2726:•
2698:)
2690:.
2652:)
2644:.
2608:)
2600:.
2553:)
2533:)
2507:)
2461:)
2447:)
2432:)
2391:12
2388:−
2370:±
2361:−
2330:)
2304:)
2283:)
2248:)
1857:)
1831:Δ
1809:)
1790:)
1774:)
1705:•
1657:•
1570:)
1526:−
1505:−
1487:−
1459:−
1411:−
1372:−
1344:−
1290:−
1228:−
1122:Δ
1060:−
996:54
978:−
962:27
959:−
888:−
866:−
844:−
765:)
742:)
722:.
706:,
690:)
650:−
615:±
598:−
554:27
539:−
486:27
449:±
436:−
385:27
348:±
335:−
274:27
237:±
193:27
156:±
106:•
76:→
4395:(
4378:(
4361:(
4306:(
4288:(
4270:(
4248:(
4233:(
4212:·
4207:(
4183:(
4167:(
4150:(
4133:(
4113:(
4090:(
4069:(
4034:(
4015:3
3999:(
3980:(
3961:(
3892:(
3869:(
3840:(
3830:?
3813:?
3788:(
3744:(
3715:3
3684:u
3681:3
3677:p
3650:3
3620:3
3617:a
3609:u
3606:+
3600:u
3597:3
3593:p
3585:=
3582:x
3498:(
3482:x
3460:3
3447:3
3434:3
3431:a
3423:=
3420:x
3400:c
3397:+
3394:x
3391:b
3388:+
3383:2
3379:x
3375:a
3372:+
3367:3
3363:x
3291:3
3286:2
3278:3
3273:)
3269:b
3266:3
3258:2
3254:a
3249:(
3244:4
3236:2
3231:)
3227:c
3221:+
3218:b
3215:a
3212:9
3204:3
3200:a
3196:2
3192:(
3182:c
3176:+
3173:b
3170:a
3167:9
3159:3
3155:a
3151:2
3142:3
3139:1
3120:3
3115:2
3107:3
3102:)
3098:b
3095:3
3087:2
3083:a
3078:(
3073:4
3065:2
3060:)
3056:c
3050:+
3047:b
3044:a
3041:9
3033:3
3029:a
3025:2
3021:(
3014:+
3011:c
3005:+
3002:b
2999:a
2996:9
2988:3
2984:a
2980:2
2971:3
2968:1
2950:3
2947:a
2935:=
2930:1
2926:x
2888:(
2873:(
2852:(
2828:(
2806:(
2783:(
2746:(
2722:(
2694:(
2648:(
2633:9
2629:7
2604:(
2549:(
2529:(
2503:(
2457:(
2443:(
2428:(
2409:a
2406:6
2397:c
2394:a
2383:2
2379:b
2375:4
2367:b
2364:2
2355:=
2352:x
2326:(
2300:(
2279:(
2244:(
1853:(
1837:0
1805:(
1786:(
1770:(
1709:)
1701:(
1696:—
1661:)
1653:(
1566:(
1538:.
1535:i
1532:)
1529:t
1523:s
1520:(
1515:2
1511:3
1499:a
1496:3
1492:b
1484:)
1481:t
1478:+
1475:s
1472:(
1467:2
1464:1
1456:=
1451:3
1447:x
1423:,
1420:i
1417:)
1414:t
1408:s
1405:(
1400:2
1396:3
1390:+
1384:a
1381:3
1377:b
1369:)
1366:t
1363:+
1360:s
1357:(
1352:2
1349:1
1341:=
1336:2
1332:x
1308:,
1302:a
1299:3
1295:b
1287:t
1284:+
1281:s
1278:=
1273:1
1269:x
1242:.
1236:3
1231:r
1225:q
1219:=
1216:t
1187:3
1182:r
1179:+
1176:q
1170:=
1167:s
1134:4
1130:a
1116:=
1109:2
1105:q
1101:+
1096:3
1091:)
1083:2
1079:a
1075:9
1068:2
1064:b
1057:c
1054:a
1051:3
1045:(
1038:=
1035:r
1004:3
1000:a
989:3
985:b
981:2
975:d
970:2
966:a
956:c
953:b
950:a
947:9
941:=
938:q
910:,
907:0
904:=
901:)
896:3
892:x
885:x
882:(
879:)
874:2
870:x
863:x
860:(
857:)
852:1
848:x
841:x
838:(
835:a
832:=
829:d
826:+
823:x
820:c
817:+
812:2
808:x
804:b
801:+
796:3
792:x
788:a
761:(
755:q
751:q
738:(
732:t
728:s
724:ρ
720:ρ
716:q
712:q
708:ρ
686:(
658:A
655:C
645:2
640:)
634:A
631:2
627:B
622:(
609:A
606:2
601:B
592:=
587:3
583:y
562:0
559:=
549:3
545:p
534:3
530:y
526:q
523:+
518:6
514:y
481:3
477:p
471:+
466:4
461:2
457:q
444:2
441:q
433:=
428:3
424:u
394:3
380:3
376:p
370:+
365:4
360:2
356:q
343:2
340:q
329:=
326:u
300:)
297:4
294:(
289:.
283:3
269:3
265:p
259:+
254:4
249:2
245:q
232:2
229:q
221:=
218:u
188:3
184:p
178:+
173:4
168:2
164:q
151:2
148:q
143:=
138:3
134:u
110:)
102:(
50:.
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