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:

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:

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