When does this quadratic with an exponential coefficient have a solution?

Question

I have an equation of the form

$$ x^2(\exp(ax)-1) - bx + c = 0 $$

where $a,b,c>0$.

It appears not to have an analytical solution, but is it possible to identify the conditions under which at least one real solution will exist?

Answer 1

The limit as $x\to -\infty$ is $-\infty$, as the exponential term tends to $0$, so the coefficient of the quadratic term is between $-1$ and $-1/2$ for all sufficiently negative $x$. The limit as $x\to \infty$ is obviously $\infty$. The intermediate value theorem then guarantees a solution.