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x^((-2)*x)+10=0 equation

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Numerical solution:

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The solution

You have entered [src] 
 -2*x         
x     + 10 = 0
$$10 + x^{- 2 x} = 0$$
Simplify
The graph
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
                                / /  log(10)   pi*I\\        / /  log(10)   pi*I\\                             
                              re|W|- ------- + ----||      re|W|- ------- + ----||                             
   /  / /  log(10)   pi*I\\\    \ \     2       2  //        \ \     2       2  //    /  / /  log(10)   pi*I\\\
cos|im|W|- ------- + ----|||*e                        + I*e                       *sin|im|W|- ------- + ----|||
   \  \ \     2       2  ///                                                          \  \ \     2       2  ///
$$e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \cos{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)} + i e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \sin{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)}$$ 
=
                                / /  log(10)   pi*I\\        / /  log(10)   pi*I\\                             
                              re|W|- ------- + ----||      re|W|- ------- + ----||                             
   /  / /  log(10)   pi*I\\\    \ \     2       2  //        \ \     2       2  //    /  / /  log(10)   pi*I\\\
cos|im|W|- ------- + ----|||*e                        + I*e                       *sin|im|W|- ------- + ----|||
   \  \ \     2       2  ///                                                          \  \ \     2       2  ///
$$e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \cos{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)} + i e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \sin{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)}$$ 
product
                                / /  log(10)   pi*I\\        / /  log(10)   pi*I\\                             
                              re|W|- ------- + ----||      re|W|- ------- + ----||                             
   /  / /  log(10)   pi*I\\\    \ \     2       2  //        \ \     2       2  //    /  / /  log(10)   pi*I\\\
cos|im|W|- ------- + ----|||*e                        + I*e                       *sin|im|W|- ------- + ----|||
   \  \ \     2       2  ///                                                          \  \ \     2       2  ///
$$e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \cos{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)} + i e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \sin{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)}$$ 
=
     / /  log(10)   pi*I\\     / /  log(10)   pi*I\\
 I*im|W|- ------- + ----|| + re|W|- ------- + ----||
     \ \     2       2  //     \ \     2       2  //
e
$$e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} + i \operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}}$$ 
exp(i*im(LambertW(-log(10)/2 + pi*i/2)) + re(LambertW(-log(10)/2 + pi*i/2)))
Rapid solution [src] 
                                     / /  log(10)   pi*I\\        / /  log(10)   pi*I\\                             
                                   re|W|- ------- + ----||      re|W|- ------- + ----||                             
        /  / /  log(10)   pi*I\\\    \ \     2       2  //        \ \     2       2  //    /  / /  log(10)   pi*I\\\
x1 = cos|im|W|- ------- + ----|||*e                        + I*e                       *sin|im|W|- ------- + ----|||
        \  \ \     2       2  ///                                                          \  \ \     2       2  ///
$$x_{1} = e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \cos{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)} + i e^{\operatorname{re}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)}} \sin{\left(\operatorname{im}{\left(W\left(- \frac{\log{\left(10 \right)}}{2} + \frac{i \pi}{2}\right)\right)} \right)}$$ 
x1 = exp(re(LambertW(-log(10)/2 + i*pi/2)))*cos(im(LambertW(-log(10)/2 + i*pi/2))) + i*exp(re(LambertW(-log(10)/2 + i*pi/2)))*sin(im(LambertW(-log(10)/2 + i*pi/2)))
Numerical answer [src] 
x1 = -1.56678401132905 - 0.14465117780799*i
x1 = -1.56678401132905 - 0.14465117780799*i
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