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September 20th, 2018, 02:09 PM   #11
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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.
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September 20th, 2018, 06:13 PM   #12
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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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