A bit of an exponential pickle.

Quote:

Originally Posted by VonNemo19

$x^2+e^{-x}=8$

Is there an analytic solution? If not, why so?

$e^xx^2+1=8e^x\Rightarrow{e}^x=\frac{-1}{x^2-8}$ ?

All this tells me is that $|x|>\sqrt8$

I didn't check your calculations but from the last equation you know that $-\sqrt{8} < x < \sqrt{8}$ :

$e^x$ is positive
Therefore the denominator of the fraction must be negative

EDIT:

Probably you have noticed that your last equation can't be used to solve it for x because you have the variable on both sides of the equation.

I "solved" (using Newton's method) the equation:

$x^2+e^{-x}-8=0$

and I got 2 real approximate results: $x_1 \approx 2.817847...~\vee~x_2\approx -1.65826...$