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x^2-(sqrt(3)+(1/sqrt(3))*i)-1=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 2       ___     I          
x  + - \/ 3  - ----- - 1 = 0
                 ___        
               \/ 3
(x2+(−3−i3))−1=0\left(x^{2} + \left(- \sqrt{3} - \frac{i}{\sqrt{3}}\right)\right) - 1 = 0
Simplificar
Solución detallada
Abramos la expresión en la ecuación
(x2+(−3−i3))−1=0\left(x^{2} + \left(- \sqrt{3} - \frac{i}{\sqrt{3}}\right)\right) - 1 = 0
Obtenemos la ecuación cuadrática
x2−3−1−3i3=0x^{2} - \sqrt{3} - 1 - \frac{\sqrt{3} i}{3} = 0
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=1a = 1
b=0b = 0
c=−3−1−3i3c = - \sqrt{3} - 1 - \frac{\sqrt{3} i}{3}
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-1 - sqrt(3) - i*sqrt(3)/3) = 4 + 4*sqrt(3) + 4*i*sqrt(3)/3

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=4+43+43i32x_{1} = \frac{\sqrt{4 + 4 \sqrt{3} + \frac{4 \sqrt{3} i}{3}}}{2}
x2=−4+43+43i32x_{2} = - \frac{\sqrt{4 + 4 \sqrt{3} + \frac{4 \sqrt{3} i}{3}}}{2}
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=−3−1−3i3q = - \sqrt{3} - 1 - \frac{\sqrt{3} i}{3}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=−3−1−3i3x_{1} x_{2} = - \sqrt{3} - 1 - \frac{\sqrt{3} i}{3}
Gráfica
Respuesta rápida [src] 
                                    /    /      ___  \\                                  /    /      ___  \\
                                    |    |  3*\/ 3   ||                                  |    |  3*\/ 3   ||
           _____________________    |atan|-----------||         _____________________    |atan|-----------||
          /                   2     |    |        ___||        /                   2     |    |        ___||
       4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /|
       \/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------|
                                    \        2        /                                  \        2        /
x1 = - ------------------------------------------------ - --------------------------------------------------
                              3                                                   3
x1=−27+(9+93)24cos⁡(atan⁡(339+93)2)3−i27+(9+93)24sin⁡(atan⁡(339+93)2)3x_{1} = - \frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} - \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} 
                                  /    /      ___  \\                                  /    /      ___  \\
                                  |    |  3*\/ 3   ||                                  |    |  3*\/ 3   ||
         _____________________    |atan|-----------||         _____________________    |atan|-----------||
        /                   2     |    |        ___||        /                   2     |    |        ___||
     4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /|
     \/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------|
                                  \        2        /                                  \        2        /
x2 = ------------------------------------------------ + --------------------------------------------------
                            3                                                   3
x2=27+(9+93)24cos⁡(atan⁡(339+93)2)3+i27+(9+93)24sin⁡(atan⁡(339+93)2)3x_{2} = \frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} + \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} 
x2 = (27 + (9 + 9*sqrt(3))^2)^(1/4)*cos(atan(3*sqrt(3)/(9 + 9*sqrt(3)))/2)/3 + i*(27 + (9 + 9*sqrt(3))^2)^(1/4)*sin(atan(3*sqrt(3)/(9 + 9*sqrt(3)))/2)/3
Suma y producto de raíces [src] 
suma
                               /    /      ___  \\                                  /    /      ___  \\                                /    /      ___  \\                                  /    /      ___  \\
                               |    |  3*\/ 3   ||                                  |    |  3*\/ 3   ||                                |    |  3*\/ 3   ||                                  |    |  3*\/ 3   ||
      _____________________    |atan|-----------||         _____________________    |atan|-----------||       _____________________    |atan|-----------||         _____________________    |atan|-----------||
     /                   2     |    |        ___||        /                   2     |    |        ___||      /                   2     |    |        ___||        /                   2     |    |        ___||
  4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /|   4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /|
  \/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------|   \/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------|
                               \        2        /                                  \        2        /                                \        2        /                                  \        2        /
- ------------------------------------------------ - -------------------------------------------------- + ------------------------------------------------ + --------------------------------------------------
                         3                                                   3                                                   3                                                   3
(−27+(9+93)24cos⁡(atan⁡(339+93)2)3−i27+(9+93)24sin⁡(atan⁡(339+93)2)3)+(27+(9+93)24cos⁡(atan⁡(339+93)2)3+i27+(9+93)24sin⁡(atan⁡(339+93)2)3)\left(- \frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} - \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3}\right) + \left(\frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} + \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3}\right) 
=
0
00 
producto
/                               /    /      ___  \\                                  /    /      ___  \\\ /                             /    /      ___  \\                                  /    /      ___  \\\
|                               |    |  3*\/ 3   ||                                  |    |  3*\/ 3   ||| |                             |    |  3*\/ 3   ||                                  |    |  3*\/ 3   |||
|      _____________________    |atan|-----------||         _____________________    |atan|-----------||| |    _____________________    |atan|-----------||         _____________________    |atan|-----------|||
|     /                   2     |    |        ___||        /                   2     |    |        ___||| |   /                   2     |    |        ___||        /                   2     |    |        ___|||
|  4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /|| |4 /       /        ___\      |    \9 + 9*\/ 3 /|     4 /       /        ___\      |    \9 + 9*\/ 3 /||
|  \/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------|| |\/   27 + \9 + 9*\/ 3 /  *cos|-----------------|   I*\/   27 + \9 + 9*\/ 3 /  *sin|-----------------||
|                               \        2        /                                  \        2        /| |                             \        2        /                                  \        2        /|
|- ------------------------------------------------ - --------------------------------------------------|*|------------------------------------------------ + --------------------------------------------------|
\                         3                                                   3                         / \                       3                                                   3                         /
(−27+(9+93)24cos⁡(atan⁡(339+93)2)3−i27+(9+93)24sin⁡(atan⁡(339+93)2)3)(27+(9+93)24cos⁡(atan⁡(339+93)2)3+i27+(9+93)24sin⁡(atan⁡(339+93)2)3)\left(- \frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} - \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3}\right) \left(\frac{\sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3} + \frac{i \sqrt[4]{27 + \left(9 + 9 \sqrt{3}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{3}}{9 + 9 \sqrt{3}} \right)}}{2} \right)}}{3}\right) 
=
                                 /      ___\ 
     ____________________        |1   \/ 3 | 
    /                  2   I*atan|- - -----| 
   /        /      ___\          \2     6  / 
-\/   3 + 9*\1 + \/ 3 /  *e                  
---------------------------------------------
                      3
−3+9(1+3)2eiatan⁡(12−36)3- \frac{\sqrt{3 + 9 \left(1 + \sqrt{3}\right)^{2}} e^{i \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3}}{6} \right)}}}{3} 
-sqrt(3 + 9*(1 + sqrt(3))^2)*exp(i*atan(1/2 - sqrt(3)/6))/3
Respuesta numérica [src] 
x1 = 1.66199271368357 + 0.173692178201555*i
x2 = -1.66199271368357 - 0.173692178201555*i
x2 = -1.66199271368357 - 0.173692178201555*i
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