Could somebody show me how to prove that $ ae^x=1+x+\frac{x^2}{2}$ has only one real root for $a>0$? All I know so far is that the equation has a root because $1+x+\frac{x^2}{2}>0$ for all real x.
There can't be two solutions, because the quadratic function is always above x axis, and so is $e^x$. When I drew a graph of those two functions I found that the only point of intersection is 0. I wish I could include the graph here but since I'm new here, my reputation is too low and I can't.