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2*(4+x^(-3))=0 equation

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Numerical solution:

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The solution

You have entered [src] 
  /    1 \    
2*|4 + --| = 0
  |     3|    
  \    x /
2(4+1x3)=02 \left(4 + \frac{1}{x^{3}}\right) = 0
Simplify
Detail solution
Given the equation
2(4+1x3)=02 \left(4 + \frac{1}{x^{3}}\right) = 0
Because equation degree is equal to = -3 - does not contain even numbers in the numerator, then
the equation has single real root.
Get the root -3-th degree of the equation sides:
We get:
1231x33=1−83\frac{1}{\sqrt[3]{2} \sqrt[3]{\frac{1}{x^{3}}}} = \frac{1}{\sqrt[3]{-8}}
or
223x2=−(−1)232\frac{2^{\frac{2}{3}} x}{2} = - \frac{\left(-1\right)^{\frac{2}{3}}}{2}
Expand brackets in the left part
x*2^2/3/2 = -(-1)^(2/3)/2

Expand brackets in the right part
x*2^2/3/2 = 1^2/3/2

Divide both parts of the equation by 2^(2/3)/2
x = -(-1)^(2/3)/2 / (2^(2/3)/2)

We get the answer: x = -(-1)^(2/3)*2^(1/3)/2

All other 2 root(s) is the complex numbers.
do replacement:
z=xz = x
then the equation will be the:
1z3=−4\frac{1}{z^{3}} = -4
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
e−3ipr3=−4\frac{e^{- 3 i p}}{r^{3}} = -4
where
r=232r = \frac{\sqrt[3]{2}}{2}
- the magnitude of the complex number
Substitute r:
e−3ip=−1e^{- 3 i p} = -1
Using Euler’s formula, we find roots for p
−isin⁡(3p)+cos⁡(3p)=−1- i \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = -1
so
cos⁡(3p)=−1\cos{\left(3 p \right)} = -1
and
−sin⁡(3p)=0- \sin{\left(3 p \right)} = 0
then
p=−2πN3−π3p = - \frac{2 \pi N}{3} - \frac{\pi}{3}
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
z1=−232z_{1} = - \frac{\sqrt[3]{2}}{2}
z2=234−233i4z_{2} = \frac{\sqrt[3]{2}}{4} - \frac{\sqrt[3]{2} \sqrt{3} i}{4}
z3=234+233i4z_{3} = \frac{\sqrt[3]{2}}{4} + \frac{\sqrt[3]{2} \sqrt{3} i}{4}
do backward replacement
z=xz = x
x=zx = z

The final answer:
x1=−232x_{1} = - \frac{\sqrt[3]{2}}{2}
x2=234−233i4x_{2} = \frac{\sqrt[3]{2}}{4} - \frac{\sqrt[3]{2} \sqrt{3} i}{4}
x3=234+233i4x_{3} = \frac{\sqrt[3]{2}}{4} + \frac{\sqrt[3]{2} \sqrt{3} i}{4}
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.520000000000000-10000000000000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
  3 ___   3 ___     3 ___   ___   3 ___     3 ___   ___
  \/ 2    \/ 2    I*\/ 2 *\/ 3    \/ 2    I*\/ 2 *\/ 3 
- ----- + ----- - ------------- + ----- + -------------
    2       4           4           4           4
(−232+(234−233i4))+(234+233i4)\left(- \frac{\sqrt[3]{2}}{2} + \left(\frac{\sqrt[3]{2}}{4} - \frac{\sqrt[3]{2} \sqrt{3} i}{4}\right)\right) + \left(\frac{\sqrt[3]{2}}{4} + \frac{\sqrt[3]{2} \sqrt{3} i}{4}\right) 
=
0
00 
product
 3 ___  /3 ___     3 ___   ___\ /3 ___     3 ___   ___\
-\/ 2   |\/ 2    I*\/ 2 *\/ 3 | |\/ 2    I*\/ 2 *\/ 3 |
-------*|----- - -------------|*|----- + -------------|
   2    \  4           4      / \  4           4      /
−232(234−233i4)(234+233i4)- \frac{\sqrt[3]{2}}{2} \left(\frac{\sqrt[3]{2}}{4} - \frac{\sqrt[3]{2} \sqrt{3} i}{4}\right) \left(\frac{\sqrt[3]{2}}{4} + \frac{\sqrt[3]{2} \sqrt{3} i}{4}\right) 
=
-1/4
−14- \frac{1}{4} 
-1/4
Rapid solution [src] 
      3 ___ 
     -\/ 2  
x1 = -------
        2
x1=−232x_{1} = - \frac{\sqrt[3]{2}}{2} 
     3 ___     3 ___   ___
     \/ 2    I*\/ 2 *\/ 3 
x2 = ----- - -------------
       4           4
x2=234−233i4x_{2} = \frac{\sqrt[3]{2}}{4} - \frac{\sqrt[3]{2} \sqrt{3} i}{4} 
     3 ___     3 ___   ___
     \/ 2    I*\/ 2 *\/ 3 
x3 = ----- + -------------
       4           4
x3=234+233i4x_{3} = \frac{\sqrt[3]{2}}{4} + \frac{\sqrt[3]{2} \sqrt{3} i}{4} 
x3 = 2^(1/3)/4 + 2^(1/3)*sqrt(3)*i/4
Numerical answer [src] 
x1 = -0.629960524947437
x2 = 0.314980262473718 - 0.545561817985861*i
x3 = 0.314980262473718 + 0.545561817985861*i
x3 = 0.314980262473718 + 0.545561817985861*i
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