What are the ways one may solve $2x+e^x=0$? What are the ways one may solve $2x+e^x=0$ ?
Thank you.

If you rewrite the equation as $e^x=-2x$, then sketch the graphs of the exponential curve $y=e^x$ and the negatively sloped line $y=-2x$, you can see there is a single point of intersection in the interval $-1\lt x\lt0$. (Actually it should be easy to see that the intersection occurs in the interval $-1/2\lt x\lt0$.)  As other answers have noted, the solution can be expressed analytically with the Lambert function, but otherwise can only be approximated numerically.
