# Express implicit equation explicitly

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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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 8 '10 at 20:48
Well, first you have to note that the vertical axis crosses the graph of your equation thrice... – J. M. Nov 8 '10 at 20:48
You have a cubic in b. There are explicit (but messy) formulas for the solution of a cubic. Try Wikipedia. – George Lowther Nov 8 '10 at 20:50
I did mean $a$ as a function of $b$. I have edited my original post. – Gus Nov 8 '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 8 '10 at 20:52

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, $$a=\frac{2}{\sqrt{3}}\sinh\left(\frac13\sinh^{-1}\left(3\sqrt{3}b\right)\right)-b.$$