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Can I find an algebraic solution for the equation below? Thank you.

$$ x+e^{x}(x+a)=b $$

GlenLee
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    More likely, there is no elementary approach for this type of problem. You can use Newton's method, which determines the approximated solutions of $x$. Other than that, I don't think there is a clean way to determine the answer. – NasuSama Mar 21 '14 at 01:58
  • @NasuSama Thank you so much! – GlenLee Apr 05 '14 at 05:22
  • The solution can be expressed in terms of Lambert W function: see https://math.stackexchange.com/a/2545838 – Jairo Bochi Feb 20 '24 at 08:18

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The equation $x + e^x (x + a) = b$ is transcendental, and, unfortunately, in this case there is no closed formula for $x$. This equation can be written as $$e^x = \frac{b - x}{x + a},$$ so it should be clear that one cannot solve it analytically.

glebovg
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As already said by other comments and answers, the equation is transcendental and there is no closed formula for the solution. Only numerical methods (such as Newton) would provide a solution.

By the way, the equation can look "nicer" if you define $x+a=y$.

Claude Leibovici
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