Basically I am trying to solve
$$ 4-0.5x^2-e^{-0.5x}=0.5x$$
The LHS is some sort of curve and RHS is a straight line.
I suppose I need to come to some sort of pq-formula term. But I never get a quadratic term and a linear term with this. Is there any simple way to solving this?
Rearrange so you have a quadratic on one side and the exponential on the other. You can see that there are exactly two points of intersection because you know that the quadratic tends to negative infinity as $|x|$ gets big, whereas the exponential is always positive, but that the quadratic is above the exponential at $x=0$.
This means that one of the points of intersection is for $x >0$, and so you can know that you can estimate that one pretty accurately by ignoring the exponential altogether (it becomes negligible very fast). Solving the quadratic you get an approximate solution that is indeed pretty good, $x$ about $5.24$, about $1$ percent error.
This is not an equation you want to approach analytically if you can help it. Maybe with a graphing calculator!