Let $a$, $b$, and $c$ be positive numbers. Consider the cubic equation $x^2(ax+b) = c$. Are there any useful bounds (upper and lower) I can put on the unique root of this equation for $x>0$? For example, we know that $x^\* <(c/a)^{1/3}$ and $x^\* < \sqrt{c/b}$. This equation can be solved explicitly, but that expression is rather complicated and I'm just looking to find an interval that contains a solution. Thanks!



Approximate roots of a cubic
Note that, since your $f(x):=x^2(ax+b)-c$ is convex on $[0,+\infty)$ the [Newton's iteration](http://en.wikipedia.org/wiki/Newton_method) with initial point $x\_0>0$ produces a sequence $x\_n$ which is decreasing for $n\ge1$ and converges to $x^\*$. 
From the equation, $y\_n:=c/ax\_n^2\, -\, b/a$ also converges to $x^\*$, but increasing. This gives you an interval $[y\_n, x\_n]$ whose endpoints are rational functions of the coefficients (and as small as you wish, for large enough $n$).


