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The implicit expression $(b-a)=(a+b)^3$ looks like it could be written explicitly for $a$ as a function of $b$. The only region of interest is for $a,b>0$ Here is what the plot looks like:

alt text

Michael Chen
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Gus
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    You cannot express $b$ as an explicit function of $a$, even on the first quadrant, because $b$ is not a function of $a$: the portion of the graph does not pass the vertical line test. On the other hand $a$ does seem like it would be a function of $b$. – Arturo Magidin Nov 08 '10 at 20:48
  • Well, first you have to note that the vertical axis crosses the graph of your equation thrice... – J. M. ain't a mathematician Nov 08 '10 at 20:48
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    You have a cubic in b. There are explicit (but messy) formulas for the solution of a cubic. Try Wikipedia. – George Lowther Nov 08 '10 at 20:50
  • I did mean $a$ as a function of $b$. I have edited my original post. – Gus Nov 08 '10 at 20:50
  • Ok, you now have a cubic with a single real root (for $a$). Again, you can use the explicit solution for the cubic. – George Lowther Nov 08 '10 at 20:52

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Writing $p=a+b$ you have the cubic equation, $$ p^3+p-2b=0. $$ This is already in "depressed cubic" form (no $p^2$ term), so it can be solved directly by standard methods. The coefficient of p is positive, so it is strictly increasing and there will be a single real root. $$ p = \sqrt[3]{\sqrt{b^2+1/27}+b}-\sqrt[3]{\sqrt{b^2+1/27}-b} $$ or, $$ a = \sqrt[3]{\sqrt{b^2+1/27}+b}-\sqrt[3]{\sqrt{b^2+1/27}-b}-b. $$ Alternatively, using the hyperbolic method, $$ a=\frac{2}{\sqrt{3}}\sinh\left(\frac13\sinh^{-1}\left(3\sqrt{3}b\right)\right)-b. $$

George Lowther
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  • +1: For including the hyperbolic method. I will delete my answer. – Aryabhata Nov 08 '10 at 21:24
  • @George Lowther thanks for your answer! – Gus Nov 08 '10 at 21:28
  • @Moron Did you delete your answer because it was incorrect or because it was more complicated? I thought it was interesting... – Gus Nov 08 '10 at 21:30
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    @Moron: I think I have to side with Gus here. I don't think there's any need to delete a good answer just because someone else has posted something similar. – George Lowther Nov 08 '10 at 21:33
  • @Gus: I deleted it because it is a subset of George's answer. If you look at the link to hyperbolic method, it has what I had written (I wasn't aware of that when I wrote my answer). @George: Your answer is a superset, not just similar :-) If it had something different, I would have kept it around. – Aryabhata Nov 08 '10 at 21:33
  • @Moron: maybe this site can borrow the custom from predecessor forums, of "summarizing replies" (adding an answer extracting the essentials of the other answers) rather than deleting any of them. – T.. Nov 08 '10 at 21:39
  • @T..: I suppose one can edit the question itself to include the summary, after a period of time. Problem is, if new answers come up later etc, things might go out of sync. (Perhaps the software can create the summary automatically.) – Aryabhata Nov 08 '10 at 21:42
  • @Moron: certainly, but the original posters are not always qualified to summarize the replies, and having other users add substantial new material to the question can lead to the impression of "others speaking for the questioner". So in many or most cases I think a separate answer would be better. I don't think the AI-ish autosummaries are possible now, but a separate field in the Q&A for summary of replies (i.e., summary in a different location from the question or the answers) would help structure the discussions more efficiently. – T.. Nov 08 '10 at 21:46
  • @T..: That answer has to be 'ticked' by the OP or voted high enough to come to the top etc. If the software supported having a summary answer (not necessarily AI), that would be ideal. – Aryabhata Nov 08 '10 at 21:50
  • @Moron: I would feature-request a summary field if there weren't other higher priority items (such as a field for recording SOURCEs of posted material). To the extent the site relies on reputation points -- and I think the reliance should be reduced in favor of a more egalitarian model -- people should be able to acquire those by posting summaries. Users who are less able to post a stream of fast answers could accumulate points by organizing the information contributed by others. – T.. Nov 08 '10 at 21:54
  • @T..: Yeah, even though it would be a useful feature to have, I don't really see an absolute need for it. Also, even if it existed, I am not sure how many questions will actually get a summary written for them (my guess is a very low %). Anyway, I guess George is being pinged by all of these comments! – Aryabhata Nov 08 '10 at 22:06
  • Sorry for the pings, I am essentially agreeing with George that addition rather than deletion of correct material is (usually) the way to go. Redundancy is not necessarily a bad thing, one can see how different people structure similar arguments, and it increases the confidence that an answer is correct. – T.. Nov 08 '10 at 22:12
  • @T..: In this case, the difference was really too minor to be of any benefit. IMO, having a lot of answers which differ only in minor details is a problem. I believe that unless something new (with respect to existing answers) is to be learned by adding an answer, don't add it. As to the confidence of correctness, the votes + comments left should be enough I suppose. – Aryabhata Nov 08 '10 at 22:18