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Equation:
Equation -2*(-2-y)-y-2=0
Equation k^2+k=0
Equation 0*x=10
Equation (a-2)(b+5)=(a-2)b+(a-2)5
Express {x} in terms of y where:
-10*x-9*y=4
-8*x+18*y=2
-4*x-9*y=4
17*x-5*y=15
Factor polynomial:
x^8+1
Identical expressions
x^ eight + one = zero
x to the power of 8 plus 1 equally 0
x to the power of eight plus one equally zero
x8+1=0
x⁸+1=0
x^8+1=O
Similar expressions
x^8-1=0

x^8+1=0 equation

The teacher will be very surprised to see your correct solution 😉

The solution

You have entered [src]

 8        
x  + 1 = 0

x^{8} + 1 = 0

Simplify

Detail solution

Given the equation

x^{8} + 1 = 0

Because equation degree is equal to = 8 and the free term = -1 < 0,
so the real solutions of the equation d'not exist

All other 8 root(s) is the complex numbers.
do replacement:

z = x

then the equation will be the:

z^{8} = -1

Any complex number can presented so:

z = r e^{i p}

substitute to the equation

r^{8} e^{8 i p} = -1

where

r = 1

- the magnitude of the complex number
Substitute r:

e^{8 i p} = -1

Using Euler’s formula, we find roots for p

i \sin{\left(8 p \right)} + \cos{\left(8 p \right)} = -1

\cos{\left(8 p \right)} = -1

and

\sin{\left(8 p \right)} = 0

then

p = \frac{\pi N}{4} + \frac{\pi}{8}

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:

z_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

z_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

z_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

z_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

z_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}

z_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}

z_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}

z_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}

do backward replacement

z = x

x = z

The final answer:

x_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

x_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

x_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

x_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

x_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}

x_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}

x_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}

x_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}

The graph

Plot the graph f1(x)

Plot the graph f2(x)

Rapid solution [src]

            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
x1 = -   /   - - -----  + I*  /   - + ----- 
       \/    2     4        \/    2     4

x_{1} = - \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
x2 =   /   - - -----  - I*  /   - + ----- 
     \/    2     4        \/    2     4

x_{2} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}

            ___________          ___________
           /       ___          /       ___ 
          /  1   \/ 2          /  1   \/ 2  
x3 = -   /   - + -----  - I*  /   - - ----- 
       \/    2     4        \/    2     4

x_{3} = - \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

          ___________          ___________
         /       ___          /       ___ 
        /  1   \/ 2          /  1   \/ 2  
x4 =   /   - + -----  + I*  /   - - ----- 
     \/    2     4        \/    2     4

x_{4} = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
x5 = I*|---------------------- + ----------------------| + ---------------------- - ----------------------
       \          2                        2           /             2                        2

x_{5} = - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
x6 = I*|---------------------- - ----------------------| + ---------------------- + ----------------------
       \          2                        2           /             2                        2

x_{6} = \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)

       /           ___________              ___________\              ___________              ___________
       |          /       ___              /       ___ |             /       ___              /       ___ 
       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
x7 = I*|---------------------- - ----------------------| - ---------------------- - ----------------------
       \          2                        2           /             2                        2

x_{7} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)

       /             ___________              ___________\              ___________              ___________
       |            /       ___              /       ___ |             /       ___              /       ___ 
       |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
       |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
       |        \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
x8 = I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------
       \            2                        2           /             2                        2

x_{8} = - \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)

x8 = -sqrt(2)*sqrt(sqrt(2)/4 + 1/2)/2 + sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2 + i*(-sqrt(2)*sqrt(sqrt(2)/4 + 1/2)/2 - sqrt(2)*sqrt(1/2 - sqrt(2)/4)/2)

Sum and product of roots [src]

sum

                                                                                                                                                                      /           ___________              ___________\              ___________              ___________     /           ___________              ___________\              ___________              ___________     /           ___________              ___________\              ___________              ___________     /             ___________              ___________\              ___________              ___________
                                                                                                                                                                      |          /       ___              /       ___ |             /       ___              /       ___      |          /       ___              /       ___ |             /       ___              /       ___      |          /       ___              /       ___ |             /       ___              /       ___      |            /       ___              /       ___ |             /       ___              /       ___ 
       ___________          ___________        ___________          ___________          ___________          ___________        ___________          ___________     |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2       |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  
      /       ___          /       ___        /       ___          /       ___          /       ___          /       ___        /       ___          /       ___      |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - -----      |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----      |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + -----      |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- 
     /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  1   \/ 2          /  1   \/ 2          /  1   \/ 2        /  1   \/ 2          /  1   \/ 2       |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4        |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4        |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4        |        \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   
-   /   - - -----  + I*  /   - + -----  +   /   - - -----  - I*  /   - + -----  + -   /   - + -----  - I*  /   - - -----  +   /   - + -----  + I*  /   - - -----  + I*|---------------------- + ----------------------| + ---------------------- - ---------------------- + I*|---------------------- - ----------------------| + ---------------------- + ---------------------- + I*|---------------------- - ----------------------| - ---------------------- - ---------------------- + I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------
  \/    2     4        \/    2     4      \/    2     4        \/    2     4        \/    2     4        \/    2     4      \/    2     4        \/    2     4        \          2                        2           /             2                        2                \          2                        2           /             2                        2                \          2                        2           /             2                        2                \            2                        2           /             2                        2

\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) + \left(\left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) + \left(\left(\left(\left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) + \left(\left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) + \left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right)\right)\right) + \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right)\right) + \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right)\right)\right)\right)

  /           ___________              ___________\     /           ___________              ___________\     /           ___________              ___________\     /             ___________              ___________\
  |          /       ___              /       ___ |     |          /       ___              /       ___ |     |          /       ___              /       ___ |     |            /       ___              /       ___ |
  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     |    ___    /  1   \/ 2       ___    /  1   \/ 2  |
  |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |     |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |     |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |     |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |
  |      \/    2     4            \/    2     4   |     |      \/    2     4            \/    2     4   |     |      \/    2     4            \/    2     4   |     |        \/    2     4            \/    2     4   |
I*|---------------------- + ----------------------| + I*|---------------------- - ----------------------| + I*|---------------------- - ----------------------| + I*|- ---------------------- - ----------------------|
  \          2                        2           /     \          2                        2           /     \          2                        2           /     \            2                        2           /

i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right) + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right) + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)

product

                                                                                                                                                                    /  /           ___________              ___________\              ___________              ___________\ /  /           ___________              ___________\              ___________              ___________\ /  /           ___________              ___________\              ___________              ___________\ /  /             ___________              ___________\              ___________              ___________\
                                                                                                                                                                    |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |          /       ___              /       ___ |             /       ___              /       ___ | |  |            /       ___              /       ___ |             /       ___              /       ___ |
/       ___________          ___________\ /     ___________          ___________\ /       ___________          ___________\ /     ___________          ___________\ |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |  ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  | |  |    ___    /  1   \/ 2       ___    /  1   \/ 2  |     ___    /  1   \/ 2       ___    /  1   \/ 2  |
|      /       ___          /       ___ | |    /       ___          /       ___ | |      /       ___          /       ___ | |    /       ___          /       ___ | |  |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - + -----    \/ 2 *  /   - - ----- | |  |\/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- | |  |\/ 2 *  /   - + -----    \/ 2 *  /   - - ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- | |  |  \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |   \/ 2 *  /   - - -----    \/ 2 *  /   - + ----- |
|     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  1   \/ 2  | |     /  1   \/ 2          /  1   \/ 2  | |   /  1   \/ 2          /  1   \/ 2  | |  |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   | |  |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   | |  |      \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   | |  |        \/    2     4            \/    2     4   |         \/    2     4            \/    2     4   |
|-   /   - - -----  + I*  /   - + ----- |*|  /   - - -----  - I*  /   - + ----- |*|-   /   - + -----  - I*  /   - - ----- |*|  /   - + -----  + I*  /   - - ----- |*|I*|---------------------- + ----------------------| + ---------------------- - ----------------------|*|I*|---------------------- - ----------------------| + ---------------------- + ----------------------|*|I*|---------------------- - ----------------------| - ---------------------- - ----------------------|*|I*|- ---------------------- - ----------------------| + ---------------------- - ----------------------|
\  \/    2     4        \/    2     4   / \\/    2     4        \/    2     4   / \  \/    2     4        \/    2     4   / \\/    2     4        \/    2     4   / \  \          2                        2           /             2                        2           / \  \          2                        2           /             2                        2           / \  \          2                        2           /             2                        2           / \  \            2                        2           /             2                        2           /

\left(- \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} + i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(\sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} - i \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}\right) \left(- \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} - i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} + i \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}\right) \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(\frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} + \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2} + i \left(- \frac{\sqrt{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{2} - \frac{\sqrt{2} \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}}}{2}\right)\right)

1

Numerical answer [src]

x1 = -0.38268343236509 + 0.923879532511287*i

x2 = -0.38268343236509 - 0.923879532511287*i

x3 = -0.923879532511287 - 0.38268343236509*i

x4 = 0.923879532511287 - 0.38268343236509*i

x5 = -0.923879532511287 + 0.38268343236509*i

x6 = 0.38268343236509 - 0.923879532511287*i

x7 = 0.38268343236509 + 0.923879532511287*i

x8 = 0.923879532511287 + 0.38268343236509*i

x8 = 0.923879532511287 + 0.38268343236509*i

The graph

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x^8+1=0 equation

Numerical solution:

The solution