I'm studying for SOA/CAS Exam P and I have boiled down a problem to say $e^{-\lambda}\left(1+\lambda +\frac{\lambda ^2}{2}\right)=\frac23$. I was stuck after this and assumed there was a way to solve it. However the solutions say it cannot be solved algabraically and rather I should substitute each of the 5 possible choices until one of them satisfies. Which got me wondering if there is a way to solve this using some kind of other method?
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5There's is no algebraic solution, only numerical (that is, approximate) methods, like the secant method, and Newton's Method. – Gerry Myerson Jan 01 '14 at 22:36
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see: https://en.wikipedia.org/wiki/Lambert_W_function#Generalizations – IV_ Oct 31 '20 at 21:13