<s>[INST] Can we solve for $a$ and $b$ only knowing the value of $z$? We have the following equation for $z$ > 0

$ a^2b+ab^2 = z $

Only the value of $z$ is known. How can we solve this equation to get the values of $a$ and $b$?
Simplifying,

$ab(a+b) = z $ 

eg. $z=84$ 
then $a = 3$, $b=4$ 
Any approach or suggestions or trial and error method ?
 [/INST] If $a,b,z \in \mathbb{R}$ then there are an infinite number of solutions.
If we assume $a=b$ then we have $2a^3=z \Rightarrow a=b=\root 3 \of {\frac z 2}$.
But if we assume $2a=b$ then we have $6a^3=z \Rightarrow a=\root 3 \of {\frac z 6}, b=2\root 3 \of {\frac z 6}$.
And, in general, if $ka=b$ then $(k+k^2)a^3=z \Rightarrow a=\root 3 \of {\frac z {k+k^2}}, b=k\root 3 \of {\frac z {k+k^2}}$.
On the other hand if $a,b$ are constrained to be integers then you can have zero, one, two or an infinite number of solutions, depending on the value of $z$.
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