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x^5+1=0

x^5+1=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

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Solución numérica:

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Solución

Ha introducido [src] 
 5        
x  + 1 = 0
x5+1=0x^{5} + 1 = 0
Simplificar
Solución detallada
Tenemos la ecuación
x5+1=0x^{5} + 1 = 0
Ya que la potencia en la ecuación es igual a = 5 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 5 de las dos partes de la ecuación:
Obtenemos:
x55=−15\sqrt[5]{x^{5}} = \sqrt[5]{-1}
o
x=−15x = \sqrt[5]{-1}
Abrimos los paréntesis en el miembro derecho de la ecuación
x = -1^1/5

Obtenemos la respuesta: x = (-1)^(1/5)

Las demás 4 raíces son complejas.
hacemos el cambio:
z=xz = x
entonces la ecuación será así:
z5=−1z^{5} = -1
Cualquier número complejo se puede presentar que:
z=reipz = r e^{i p}
sustituimos en la ecuación
r5e5ip=−1r^{5} e^{5 i p} = -1
donde
r=1r = 1
- módulo del número complejo
Sustituyamos r:
e5ip=−1e^{5 i p} = -1
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(5p)+cos⁡(5p)=−1i \sin{\left(5 p \right)} + \cos{\left(5 p \right)} = -1
es decir
cos⁡(5p)=−1\cos{\left(5 p \right)} = -1
y
sin⁡(5p)=0\sin{\left(5 p \right)} = 0
entonces
p=2πN5+π5p = \frac{2 \pi N}{5} + \frac{\pi}{5}
donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para z
Es decir, la solución será para z:
z1=−1z_{1} = -1
z2=14+54+i58−58z_{2} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}
z3=−54+14−5i58−582−i58−582z_{3} = - \frac{\sqrt{5}}{4} + \frac{1}{4} - \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}
z4=14+58−5858+58−5i58+584−i58+584−i58−584+5i58−584z_{4} = \frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}
z5=−58−5858+58+14−i58−584+i58+584+5i58−584+5i58+584z_{5} = - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
hacemos cambio inverso
z=xz = x
x=zx = z

Entonces la respuesta definitiva es:
x1=−1x_{1} = -1
x2=14+54+i58−58x_{2} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}
x3=−54+14−5i58−582−i58−582x_{3} = - \frac{\sqrt{5}}{4} + \frac{1}{4} - \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}
x4=14+58−5858+58−5i58+584−i58+584−i58−584+5i58−584x_{4} = \frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} - \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}
x5=−58−5858+58+14−i58−584+i58+584+5i58−584+5i58+584x_{5} = - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} - \frac{i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}
Gráfica
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.5-200000200000
Trazar el gráfico f1(x)
Trazar el gráfico f2(x)
Suma y producto de raíces [src] 
suma
                   /       ___________              ___________\         /       ___________        ___________              ___________              ___________\                                             /       ___________        ___________              ___________              ___________\                                                                     
                   |      /       ___              /       ___ |         |      /       ___        /       ___              /       ___              /       ___ |                                             |      /       ___        /       ___              /       ___              /       ___ |                                                                     
                   |     /  5   \/ 5       ___    /  5   \/ 5  |         |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________         |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________                      ___________
           ___     |    /   - - -----    \/ 5 *  /   - - ----- |         |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |       /       ___      /       ___          |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |       /       ___      /       ___          ___         /       ___ 
     1   \/ 5      |  \/    8     8            \/    8     8   |   1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5     1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5     1   \/ 5         /  5   \/ 5  
-1 + - - ----- + I*|- ---------------- - ----------------------| + - + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + -----  + - + I*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + -----  + - + ----- + I*  /   - - ----- 
     4     4       \         2                     2           /   4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8      4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8      4     4       \/    8     8
(((−1+(−54+14+i(−558−582−58−582)))+(14+58−5858+58+i(−558+584−58+584−58−584+558−584)))+(−58−5858+58+14+i(−58−584+58+584+558−584+558+584)))+(14+54+i58−58)\left(\left(\left(-1 + \left(- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right)\right)\right) + \left(\frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)\right)\right) + \left(- \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right)\right) + \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) 
=
                       /       ___________              ___________\     /       ___________        ___________              ___________              ___________\     /       ___________        ___________              ___________              ___________\
                       |      /       ___              /       ___ |     |      /       ___        /       ___              /       ___              /       ___ |     |      /       ___        /       ___              /       ___              /       ___ |
       ___________     |     /  5   \/ 5       ___    /  5   \/ 5  |     |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |     |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |
      /       ___      |    /   - - -----    \/ 5 *  /   - - ----- |     |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |     |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |
     /  5   \/ 5       |  \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |
I*  /   - - -----  + I*|- ---------------- - ----------------------| + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| + I*|- ---------------- + ---------------- + ---------------------- + ----------------------|
  \/    8     8        \         2                     2           /     \         4                  4                     4                        4           /     \         4                  4                     4                        4           /
i(−558−582−58−582)+i(−558+584−58+584−58−584+558−584)+i58−58+i(−58−584+58+584+558−584+558+584)i \left(- \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right) + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right) 
producto
 /              /       ___________              ___________\\ /      /       ___________        ___________              ___________              ___________\                                    \ /      /       ___________        ___________              ___________              ___________\                                    \                                 
 |              |      /       ___              /       ___ || |      |      /       ___        /       ___              /       ___              /       ___ |                                    | |      |      /       ___        /       ___              /       ___              /       ___ |                                    |                                 
 |              |     /  5   \/ 5       ___    /  5   \/ 5  || |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________| |      |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________| /                   ___________\
 |      ___     |    /   - - -----    \/ 5 *  /   - - ----- || |      |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |       /       ___      /       ___ | |      |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |       /       ___      /       ___ | |      ___         /       ___ |
 |1   \/ 5      |  \/    8     8            \/    8     8   || |1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5  | |1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5  | |1   \/ 5         /  5   \/ 5  |
-|- - ----- + I*|- ---------------- - ----------------------||*|- + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + ----- |*|- + I*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + ----- |*|- + ----- + I*  /   - - ----- |
 \4     4       \         2                     2           // \4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8   / \4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8   / \4     4       \/    8     8   /
(14+58−5858+58+i(−558+584−58+584−58−584+558−584))(−(−54+14+i(−558−582−58−582)))(−58−5858+58+14+i(−58−584+58+584+558−584+558+584))(14+54+i58−58)\left(\frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right)\right) \left(- (- \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right))\right) \left(- \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right)\right) \left(\frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}\right) 
=
/               _______________\ /                  ____________                 \ /                /   ______________      _______________      _______________      ______________\\ /                /   _______________      ______________      ______________      _______________\\
|      ___     /           ___ | |         ___     /        ___  /  ___     ____\| |        ___     |  /          ___      /           ___      /           ___      /          ___ || |        ___     |  /           ___      /          ___      /          ___      /           ___ ||
\1 + \/ 5  + \/  -10 + 2*\/ 5  /*\-2 + 2*\/ 5  + \/  -5 + \/ 5  *\\/ 2  + \/ 10 //*\4 - 4*\/ 5  + I*\\/  10 + 2*\/ 5   + \/  50 - 10*\/ 5   + \/  50 + 10*\/ 5   - \/  10 - 2*\/ 5  //*\4 + 4*\/ 5  + I*\\/  50 - 10*\/ 5   - \/  10 - 2*\/ 5   - \/  10 + 2*\/ 5   - \/  50 + 10*\/ 5  //
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                           8192
(−2+25+−5+5(2+10))(1+5+−10+25)(4+45+i(−105+50−25+10−10−25+50−105))(−45+4+i(−10−25+25+10+50−105+105+50))8192\frac{\left(-2 + 2 \sqrt{5} + \sqrt{-5 + \sqrt{5}} \left(\sqrt{2} + \sqrt{10}\right)\right) \left(1 + \sqrt{5} + \sqrt{-10 + 2 \sqrt{5}}\right) \left(4 + 4 \sqrt{5} + i \left(- \sqrt{10 \sqrt{5} + 50} - \sqrt{2 \sqrt{5} + 10} - \sqrt{10 - 2 \sqrt{5}} + \sqrt{50 - 10 \sqrt{5}}\right)\right) \left(- 4 \sqrt{5} + 4 + i \left(- \sqrt{10 - 2 \sqrt{5}} + \sqrt{2 \sqrt{5} + 10} + \sqrt{50 - 10 \sqrt{5}} + \sqrt{10 \sqrt{5} + 50}\right)\right)}{8192} 
(1 + sqrt(5) + sqrt(-10 + 2*sqrt(5)))*(-2 + 2*sqrt(5) + sqrt(-5 + sqrt(5))*(sqrt(2) + sqrt(10)))*(4 - 4*sqrt(5) + i*(sqrt(10 + 2*sqrt(5)) + sqrt(50 - 10*sqrt(5)) + sqrt(50 + 10*sqrt(5)) - sqrt(10 - 2*sqrt(5))))*(4 + 4*sqrt(5) + i*(sqrt(50 - 10*sqrt(5)) - sqrt(10 - 2*sqrt(5)) - sqrt(10 + 2*sqrt(5)) - sqrt(50 + 10*sqrt(5))))/8192
Respuesta rápida [src] 
x1 = -1
x1=−1x_{1} = -1 
                   /       ___________              ___________\
                   |      /       ___              /       ___ |
                   |     /  5   \/ 5       ___    /  5   \/ 5  |
           ___     |    /   - - -----    \/ 5 *  /   - - ----- |
     1   \/ 5      |  \/    8     8            \/    8     8   |
x2 = - - ----- + I*|- ---------------- - ----------------------|
     4     4       \         2                     2           /
x2=−54+14+i(−558−582−58−582)x_{2} = - \frac{\sqrt{5}}{4} + \frac{1}{4} + i \left(- \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{2}\right) 
           /       ___________        ___________              ___________              ___________\                                    
           |      /       ___        /       ___              /       ___              /       ___ |                                    
           |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________
           |    /   - - -----      /   - + -----    \/ 5 *  /   - + -----    \/ 5 *  /   - - ----- |       /       ___      /       ___ 
     1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5  
x3 = - + I*|- ---------------- - ---------------- - ---------------------- + ----------------------| +   /   - - ----- *  /   - + ----- 
     4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8
x3=14+58−5858+58+i(−558+584−58+584−58−584+558−584)x_{3} = \frac{1}{4} + \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + i \left(- \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} - \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4}\right) 
           /       ___________        ___________              ___________              ___________\                                    
           |      /       ___        /       ___              /       ___              /       ___ |                                    
           |     /  5   \/ 5        /  5   \/ 5       ___    /  5   \/ 5       ___    /  5   \/ 5  |        ___________      ___________
           |    /   - - -----      /   - + -----    \/ 5 *  /   - - -----    \/ 5 *  /   - + ----- |       /       ___      /       ___ 
     1     |  \/    8     8      \/    8     8            \/    8     8            \/    8     8   |      /  5   \/ 5      /  5   \/ 5  
x4 = - + I*|- ---------------- + ---------------- + ---------------------- + ----------------------| -   /   - - ----- *  /   - + ----- 
     4     \         4                  4                     4                        4           /   \/    8     8    \/    8     8
x4=−58−5858+58+14+i(−58−584+58+584+558−584+558+584)x_{4} = - \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} + \frac{1}{4} + i \left(- \frac{\sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\sqrt{5} \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{4}\right) 
                        ___________
           ___         /       ___ 
     1   \/ 5         /  5   \/ 5  
x5 = - + ----- + I*  /   - - ----- 
     4     4       \/    8     8
x5=14+54+i58−58x_{5} = \frac{1}{4} + \frac{\sqrt{5}}{4} + i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} 
x5 = 1/4 + sqrt(5)/4 + i*sqrt(5/8 - sqrt(5)/8)
Respuesta numérica [src] 
x1 = 0.809016994374947 + 0.587785252292473*i
x2 = -1.0
x3 = 0.809016994374947 - 0.587785252292473*i
x4 = -0.309016994374947 + 0.951056516295154*i
x5 = -0.309016994374947 - 0.951056516295154*i
x5 = -0.309016994374947 - 0.951056516295154*i
Gráfico
x^5+1=0 la ecuación
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