Is it possible to get a symbolic or numeric relation when solving a transcendental equation?

Not an answer, just a long comment.

By making small Manipulate, it shows that no real solution exist when $$a$$ gets over value of around $$-1/10$$. Hard to determine the exact value of $$a$$ but using the slider you can get very close to finding it. i.e when $$a$$ is larger than that limit, it never crosses the x-axis, so no real solution exist.

``````ClearAll(a,x);
eq = (1 - Exp(x) + (3 x^2)/2)/(2 x) - a;
Manipulate(Plot(eq /. a -> a0, {x, -3, 3}),
{{a0, 0, "a"}, -1, 1, .001, Appearance -> "Labeled"},
TrackedSymbols :> {a0}
)
``````

I think this might explain why Solve could not do it in addition to the other comments above.