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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:

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    Ok, you now have$a$cubic with$a$single real root (for $a$). Again, you can use the explicit solution for the cubic.2010-11-08

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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. $

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    @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.2010-11-08