September 20th, 2018, 03:09 PM  #11 
Senior Member Joined: May 2016 From: USA Posts: 1,305 Thanks: 548 
There are at two possible questions, which are quite different. One is whether a real root of f(x) exists. That does not require finding a numerical solution at all. You can show using the Mean Value Theorem that a real root exists between 0 and 1. A different question is whether you can find a numerical approximation of that root directly from the equation itself. The answer to that is also yes. I took an INTRODUCTORY course in numerical methods 50 years ago and no longer remember much. But bisection will work here though it may very well be tedious. 
September 20th, 2018, 07:13 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 20,269 Thanks: 1958 
The solution of e^x + x  3 = 0 is x = 3  W(e³), where W denotes the Lambert W function. The value of a is approximately 0.792059968430677. 

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