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x^3-8/(-1-sqrt(3)*i)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 3        8          
x  - ------------ = 0
            ___      
     -1 - \/ 3 *I
x3−8−1−3i=0x^{3} - \frac{8}{-1 - \sqrt{3} i} = 0
Simplificar
Solución detallada
Tenemos la ecuación
x3−8−1−3i=0x^{3} - \frac{8}{-1 - \sqrt{3} i} = 0
Ya que la potencia en la ecuación es igual a = 3 - no contiene número par en el numerador, entonces
la ecuación tendrá una raíz real.
Extraigamos la raíz de potencia 3 de las dos partes de la ecuación:
Obtenemos:
x33=8−1−3i3\sqrt[3]{x^{3}} = \sqrt[3]{\frac{8}{-1 - \sqrt{3} i}}
o
x=21−1−3i3x = 2 \sqrt[3]{\frac{1}{-1 - \sqrt{3} i}}
Abrimos los paréntesis en el miembro derecho de la ecuación
x = 2*1/-2*1+2*i*sqrt+2*3))^1/3

Sumamos los términos semejantes en el miembro derecho de la ecuación:
x = 2*(1/(-1 - i*sqrt(3)))^(1/3)

Obtenemos la respuesta: x = 2*(-1/(1 + i*sqrt(3)))^(1/3)

Las demás 3 raíces son complejas.
hacemos el cambio:
z=xz = x
entonces la ecuación será así:
z3=8−1−3iz^{3} = \frac{8}{-1 - \sqrt{3} i}
Cualquier número complejo se puede presentar que:
z=reipz = r e^{i p}
sustituimos en la ecuación
r3e3ip=8−1−3ir^{3} e^{3 i p} = \frac{8}{-1 - \sqrt{3} i}
donde
r=223r = 2^{\frac{2}{3}}
- módulo del número complejo
Sustituyamos r:
e3ip=2−1−3ie^{3 i p} = \frac{2}{-1 - \sqrt{3} i}
Usando la fórmula de Euler hallemos las raíces para p
isin⁡(3p)+cos⁡(3p)=2−1−3ii \sin{\left(3 p \right)} + \cos{\left(3 p \right)} = \frac{2}{-1 - \sqrt{3} i}
es decir
cos⁡(3p)=−12\cos{\left(3 p \right)} = - \frac{1}{2}
y
sin⁡(3p)=32\sin{\left(3 p \right)} = \frac{\sqrt{3}}{2}
entonces
p=2πN3−π9p = \frac{2 \pi N}{3} - \frac{\pi}{9}
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=223cos⁡(2π9)+223isin⁡(2π9)z_{1} = 2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)} + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}
z2=−223cos⁡(2π9)2+2233sin⁡(2π9)2−2233icos⁡(2π9)2−223isin⁡(2π9)2z_{2} = - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sqrt{3} i \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}}{2}
z3=−2233sin⁡(2π9)2−223cos⁡(2π9)2−223isin⁡(2π9)2+2233icos⁡(2π9)2z_{3} = - \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} i \cos{\left(\frac{2 \pi}{9} \right)}}{2}
hacemos cambio inverso
z=xz = x
x=zx = z

Entonces la respuesta definitiva es:
x1=223cos⁡(2π9)+223isin⁡(2π9)x_{1} = 2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)} + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}
x2=−223cos⁡(2π9)2+2233sin⁡(2π9)2−2233icos⁡(2π9)2−223isin⁡(2π9)2x_{2} = - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sqrt{3} i \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}}{2}
x3=−2233sin⁡(2π9)2−223cos⁡(2π9)2−223isin⁡(2π9)2+2233icos⁡(2π9)2x_{3} = - \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} i \cos{\left(\frac{2 \pi}{9} \right)}}{2}
Teorema de Cardano-Vieta
es ecuación cúbica reducida
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=0q = 0
v=dav = \frac{d}{a}
v=−8−1−3iv = - \frac{8}{-1 - \sqrt{3} i}
Fórmulas de Cardano-Vieta
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=0x_{1} + x_{2} + x_{3} = 0
x1x2+x1x3+x2x3=0x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0
x1x2x3=−8−1−3ix_{1} x_{2} x_{3} = - \frac{8}{-1 - \sqrt{3} i}
Gráfica
Suma y producto de raíces [src] 
suma
                                      /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\     /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\
                                      |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----|     |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----|
 2/3    /2*pi\      2/3    /2*pi\     |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /     |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /
2   *cos|----| + I*2   *sin|----| + I*|- -------------- - --------------------| - -------------- + -------------------- + I*|- -------------- + --------------------| - -------------- - --------------------
        \ 9  /             \ 9  /     \        2                   2          /         2                   2               \        2                   2          /         2                   2
((−223cos⁡(2π9)2+2233sin⁡(2π9)2+i(−2233cos⁡(2π9)2−223sin⁡(2π9)2))+(223cos⁡(2π9)+223isin⁡(2π9)))+(−2233sin⁡(2π9)2−223cos⁡(2π9)2+i(−223sin⁡(2π9)2+2233cos⁡(2π9)2))\left(\left(- \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2}\right)\right) + \left(2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)} + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}\right)\right) + \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2}\right)\right) 
=
  /   2/3    /2*pi\    2/3   ___    /2*pi\\     /   2/3    /2*pi\    2/3   ___    /2*pi\\                   
  |  2   *sin|----|   2   *\/ 3 *cos|----||     |  2   *sin|----|   2   *\/ 3 *cos|----||                   
  |          \ 9  /                 \ 9  /|     |          \ 9  /                 \ 9  /|      2/3    /2*pi\
I*|- -------------- + --------------------| + I*|- -------------- - --------------------| + I*2   *sin|----|
  \        2                   2          /     \        2                   2          /             \ 9  /
i(−2233cos⁡(2π9)2−223sin⁡(2π9)2)+i(−223sin⁡(2π9)2+2233cos⁡(2π9)2)+223isin⁡(2π9)i \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2}\right) + i \left(- \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2}\right) + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)} 
producto
                                    /  /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\\ /  /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\\
                                    |  |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----|| |  |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----||
/ 2/3    /2*pi\      2/3    /2*pi\\ |  |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /| |  |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /|
|2   *cos|----| + I*2   *sin|----||*|I*|- -------------- - --------------------| - -------------- + --------------------|*|I*|- -------------- + --------------------| - -------------- - --------------------|
\        \ 9  /             \ 9  // \  \        2                   2          /         2                   2          / \  \        2                   2          /         2                   2          /
(223cos⁡(2π9)+223isin⁡(2π9))(−223cos⁡(2π9)2+2233sin⁡(2π9)2+i(−2233cos⁡(2π9)2−223sin⁡(2π9)2))(−2233sin⁡(2π9)2−223cos⁡(2π9)2+i(−223sin⁡(2π9)2+2233cos⁡(2π9)2))\left(2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)} + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)}\right) \left(- \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2}\right)\right) \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2}\right)\right) 
=
  3        /2*pi\        3/2*pi\          3/2*pi\           2/2*pi\    /2*pi\
- - - 3*cos|----| + 4*cos |----| - 4*I*sin |----| + 12*I*cos |----|*sin|----|
  2        \ 9  /         \ 9  /           \ 9  /            \ 9  /    \ 9  /
−3cos⁡(2π9)−32+4cos⁡3(2π9)−4isin⁡3(2π9)+12isin⁡(2π9)cos⁡2(2π9)- 3 \cos{\left(\frac{2 \pi}{9} \right)} - \frac{3}{2} + 4 \cos^{3}{\left(\frac{2 \pi}{9} \right)} - 4 i \sin^{3}{\left(\frac{2 \pi}{9} \right)} + 12 i \sin{\left(\frac{2 \pi}{9} \right)} \cos^{2}{\left(\frac{2 \pi}{9} \right)} 
-3/2 - 3*cos(2*pi/9) + 4*cos(2*pi/9)^3 - 4*i*sin(2*pi/9)^3 + 12*i*cos(2*pi/9)^2*sin(2*pi/9)
Respuesta rápida [src] 
      2/3    /2*pi\      2/3    /2*pi\
x1 = 2   *cos|----| + I*2   *sin|----|
             \ 9  /             \ 9  /
x1=223cos⁡(2π9)+223isin⁡(2π9)x_{1} = 2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)} + 2^{\frac{2}{3}} i \sin{\left(\frac{2 \pi}{9} \right)} 
       /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\
       |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----|
       |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /
x2 = I*|- -------------- - --------------------| - -------------- + --------------------
       \        2                   2          /         2                   2
x2=−223cos⁡(2π9)2+2233sin⁡(2π9)2+i(−2233cos⁡(2π9)2−223sin⁡(2π9)2)x_{2} = - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2}\right) 
       /   2/3    /2*pi\    2/3   ___    /2*pi\\    2/3    /2*pi\    2/3   ___    /2*pi\
       |  2   *sin|----|   2   *\/ 3 *cos|----||   2   *cos|----|   2   *\/ 3 *sin|----|
       |          \ 9  /                 \ 9  /|           \ 9  /                 \ 9  /
x3 = I*|- -------------- + --------------------| - -------------- - --------------------
       \        2                   2          /         2                   2
x3=−2233sin⁡(2π9)2−223cos⁡(2π9)2+i(−223sin⁡(2π9)2+2233cos⁡(2π9)2)x_{3} = - \frac{2^{\frac{2}{3}} \sqrt{3} \sin{\left(\frac{2 \pi}{9} \right)}}{2} - \frac{2^{\frac{2}{3}} \cos{\left(\frac{2 \pi}{9} \right)}}{2} + i \left(- \frac{2^{\frac{2}{3}} \sin{\left(\frac{2 \pi}{9} \right)}}{2} + \frac{2^{\frac{2}{3}} \sqrt{3} \cos{\left(\frac{2 \pi}{9} \right)}}{2}\right) 
x3 = -2^(2/3)*sqrt(3)*sin(2*pi/9)/2 - 2^(2/3)*cos(2*pi/9)/2 + i*(-2^(2/3)*sin(2*pi/9)/2 + 2^(2/3)*sqrt(3)*cos(2*pi/9)/2)
Respuesta numérica [src] 
x1 = 1.21601975486146 + 1.02036172780854*i
x2 = -1.49166905476231 + 0.542923135309481*i
x3 = 0.275649299900846 - 1.56328486311802*i
x3 = 0.275649299900846 - 1.56328486311802*i
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