# Solving $y = xc^x + x + 1$, where c is a constant

How do you solve $xc^x + x + 1 = 0$ for $x$, where $c$ is a constant?

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Doesn't look like something you can solve neatly, or even use the Lambert function on... – J. M. Oct 23 '11 at 8:41
small comment on existence..for sufficiently large negative values of $x$, $y$ is negative and for some positive values of $x$, $y$ is positive, since the function is continuous it seems to have a solution... – Dinesh Oct 23 '11 at 8:41
@Dinesh: I think your statements require positive $c$. – Henry Oct 23 '11 at 9:22
@Henry yes..I missed that. – Dinesh Oct 23 '11 at 9:26
Title $\ne$ body. Do you want $y$ in there, or do you want zero? Please decide and edit accordingly. – Gerry Myerson Oct 23 '11 at 11:35

## 1 Answer

There doesn't exist a simple formula, in terms of elementary functions, for finding the roots of functions of this form.

If you need the root for particular $c$ (and $c\neq 0, 1$) you will need to approximate the root, for instance by plotting the function and looking for $x$-intercepts, or asking Wolfram Alpha. There also exist advanced numerical techniques, such as Newton's Method, that you could use to very accurately approximate roots of functions yourself.

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Are you saying Newton's Method is more advanced than Wolfram Alpha? – Robert Israel Oct 23 '11 at 9:12
No. But if you're a precalculus student, using Alpha as a black box to plot and find the root is a reasonable approach to the problem, while understanding and implementing Newton's method is likely not (unless your precalc class was a lot more interesting than mine!) I've tried to clean up the wording of the answer. – user7530 Oct 23 '11 at 9:37