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(x-1)^3=-8

(x-1)^3=-8 equation

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Numerical solution:

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The solution

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       3     
(x - 1)  = -8
(x−1)3=−8\left(x - 1\right)^{3} = -8
Simplify
Detail solution
Given the equation
(x−1)3=−8\left(x - 1\right)^{3} = -8
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:
(x−1)33=−83\sqrt[3]{\left(x - 1\right)^{3}} = \sqrt[3]{-8}
or
x−1=2−13x - 1 = 2 \sqrt[3]{-1}
Expand brackets in the right part
-1 + x = -2*1^1/3

Move free summands (without x)
from left part to right part, we given:
x=1+2−13x = 1 + 2 \sqrt[3]{-1}
We get the answer: x = 1 + 2*(-1)^(1/3)

All other 2 root(s) is the complex numbers.
do replacement:
z=x−1z = x - 1
then the equation will be the:
z3=−8z^{3} = -8
Any complex number can presented so:
z=reipz = r e^{i p}
substitute to the equation
r3e3ip=−8r^{3} e^{3 i p} = -8
where
r=2r = 2
- the magnitude of the complex number
Substitute r:
e3ip=−1e^{3 i p} = -1
Using Euler’s formula, we find roots for p
isin⁡(3p)+cos⁡(3p)=−1i \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=−2z_{1} = -2
z2=1−3iz_{2} = 1 - \sqrt{3} i
z3=1+3iz_{3} = 1 + \sqrt{3} i
do backward replacement
z=x−1z = x - 1
x=z+1x = z + 1

The final answer:
x1=−1x_{1} = -1
x2=2−3ix_{2} = 2 - \sqrt{3} i
x3=2+3ix_{3} = 2 + \sqrt{3} i
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-25002500
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1 
             ___
x2 = 2 - I*\/ 3
x2=2−3ix_{2} = 2 - \sqrt{3} i 
             ___
x3 = 2 + I*\/ 3
x3=2+3ix_{3} = 2 + \sqrt{3} i 
x3 = 2 + sqrt(3)*i
Sum and product of roots [src] 
sum
             ___           ___
-1 + 2 - I*\/ 3  + 2 + I*\/ 3
(−1+(2−3i))+(2+3i)\left(-1 + \left(2 - \sqrt{3} i\right)\right) + \left(2 + \sqrt{3} i\right) 
=
3
33 
product
 /        ___\ /        ___\
-\2 - I*\/ 3 /*\2 + I*\/ 3 /
−(2−3i)(2+3i)- (2 - \sqrt{3} i) \left(2 + \sqrt{3} i\right) 
=
-7
−7-7 
-7
Numerical answer [src] 
x1 = 2.0 - 1.73205080756888*i
x2 = 2.0 + 1.73205080756888*i
x3 = -1.0
x3 = -1.0
The graph
(x-1)^3=-8 equation
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