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z^2+1+sqrt(3*i)=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
 2         _____    
z  + 1 + \/ 3*I  = 0
(z2+1)+3i=0\left(z^{2} + 1\right) + \sqrt{3 i} = 0
Simplificar
Solución detallada
Abramos la expresión en la ecuación
(z2+1)+3i=0\left(z^{2} + 1\right) + \sqrt{3 i} = 0
Obtenemos la ecuación cuadrática
z2+1+3i=0z^{2} + 1 + \sqrt{3} \sqrt{i} = 0
Es la ecuación de la forma
a*z^2 + b*z + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
z1=D−b2az_{1} = \frac{\sqrt{D} - b}{2 a}
z2=−D−b2az_{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=1+3ic = 1 + \sqrt{3} \sqrt{i}
, entonces
D = b^2 - 4 * a * c = 

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

La ecuación tiene dos raíces.
z1 = (-b + sqrt(D)) / (2*a)

z2 = (-b - sqrt(D)) / (2*a)

o
z1=−4−43i2z_{1} = \frac{\sqrt{-4 - 4 \sqrt{3} \sqrt{i}}}{2}
z2=−−4−43i2z_{2} = - \frac{\sqrt{-4 - 4 \sqrt{3} \sqrt{i}}}{2}
Teorema de Cardano-Vieta
es ecuación cuadrática reducida
pz+q+z2=0p z + q + z^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=1+3iq = 1 + \sqrt{3} \sqrt{i}
Fórmulas de Cardano-Vieta
z1+z2=−pz_{1} + z_{2} = - p
z1z2=qz_{1} z_{2} = q
z1+z2=0z_{1} + z_{2} = 0
z1z2=1+3iz_{1} z_{2} = 1 + \sqrt{3} \sqrt{i}
Gráfica
Respuesta rápida [src] 
                                  /    /      ___     \\                                  /    /      ___     \\
                                  |    |    \/ 6      ||                                  |    |    \/ 6      ||
                                  |atan|--------------||                                  |atan|--------------||
           ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||
          /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||
         /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||
        /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|
z1 = 4 /    - + |-1 - -----|  *sin|--------------------| + I*4 /    - + |-1 - -----|  *cos|--------------------|
     \/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          /
z1=32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)+i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2)z_{1} = \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} + i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} 
                                    /    /      ___     \\                                  /    /      ___     \\
                                    |    |    \/ 6      ||                                  |    |    \/ 6      ||
                                    |atan|--------------||                                  |atan|--------------||
             ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||
            /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||
           /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||
          /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|
z2 = - 4 /    - + |-1 - -----|  *sin|--------------------| - I*4 /    - + |-1 - -----|  *cos|--------------------|
       \/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          /
z2=−32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)−i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2)z_{2} = - \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} - i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} 
z2 = -(3/2 + (-sqrt(6)/2 - 1)^2)^(1/4)*sin(atan(sqrt(6)/(2*(-sqrt(6)/2 - 1)))/2) - i*(3/2 + (-sqrt(6)/2 - 1)^2)^(1/4)*cos(atan(sqrt(6)/(2*(-sqrt(6)/2 - 1)))/2)
Suma y producto de raíces [src] 
suma
                             /    /      ___     \\                                  /    /      ___     \\                                  /    /      ___     \\                                  /    /      ___     \\
                             |    |    \/ 6      ||                                  |    |    \/ 6      ||                                  |    |    \/ 6      ||                                  |    |    \/ 6      ||
                             |atan|--------------||                                  |atan|--------------||                                  |atan|--------------||                                  |atan|--------------||
      ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||
     /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||
    /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||
   /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|
4 /    - + |-1 - -----|  *sin|--------------------| + I*4 /    - + |-1 - -----|  *cos|--------------------| + - 4 /    - + |-1 - -----|  *sin|--------------------| - I*4 /    - + |-1 - -----|  *cos|--------------------|
\/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          /
(−32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)−i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2))+(32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)+i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2))\left(- \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} - i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)}\right) + \left(\sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} + i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)}\right) 
=
0
00 
producto
/                             /    /      ___     \\                                  /    /      ___     \\\ /                               /    /      ___     \\                                  /    /      ___     \\\
|                             |    |    \/ 6      ||                                  |    |    \/ 6      ||| |                               |    |    \/ 6      ||                                  |    |    \/ 6      |||
|                             |atan|--------------||                                  |atan|--------------||| |                               |atan|--------------||                                  |atan|--------------|||
|      ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\||| |        ___________________    |    |  /       ___\||           ___________________    |    |  /       ___\|||
|     /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 |||| |       /                 2     |    |  |     \/ 6 |||          /                 2     |    |  |     \/ 6 ||||
|    /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----|||| |      /      /       ___\      |    |2*|-1 - -----|||         /      /       ___\      |    |2*|-1 - -----||||
|   /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //|| |     /   3   |     \/ 6 |      |    \  \       2  //|        /   3   |     \/ 6 |      |    \  \       2  //||
|4 /    - + |-1 - -----|  *sin|--------------------| + I*4 /    - + |-1 - -----|  *cos|--------------------||*|- 4 /    - + |-1 - -----|  *sin|--------------------| - I*4 /    - + |-1 - -----|  *cos|--------------------||
\\/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          // \  \/     2   \       2  /      \         2          /     \/     2   \       2  /      \         2          //
(−32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)−i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2))(32+(−62−1)24sin⁡(atan⁡(62(−62−1))2)+i32+(−62−1)24cos⁡(atan⁡(62(−62−1))2))\left(- \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} - i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)}\right) \left(\sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)} + i \sqrt[4]{\frac{3}{2} + \left(- \frac{\sqrt{6}}{2} - 1\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{6}}{2 \left(- \frac{\sqrt{6}}{2} - 1\right)} \right)}}{2} \right)}\right) 
=
     __________________                                                  2 
    /                2  /       /    /      ___\\      /    /      ___\\\  
   /      /      ___\   |       |atan\3 - \/ 6 /|      |atan\3 - \/ 6 /||  
-\/   6 + \2 + \/ 6 /  *|- I*cos|---------------| + sin|---------------||  
                        \       \       2       /      \       2       //  
---------------------------------------------------------------------------
                                     2
−6+(2+6)2(sin⁡(atan⁡(3−6)2)−icos⁡(atan⁡(3−6)2))22- \frac{\sqrt{6 + \left(2 + \sqrt{6}\right)^{2}} \left(\sin{\left(\frac{\operatorname{atan}{\left(3 - \sqrt{6} \right)}}{2} \right)} - i \cos{\left(\frac{\operatorname{atan}{\left(3 - \sqrt{6} \right)}}{2} \right)}\right)^{2}}{2} 
-sqrt(6 + (2 + sqrt(6))^2)*(-i*cos(atan(3 - sqrt(6))/2) + sin(atan(3 - sqrt(6))/2))^2/2
Respuesta numérica [src] 
z1 = 0.396761697792363 - 1.54342629115441*i
z2 = -0.396761697792363 + 1.54342629115441*i
z2 = -0.396761697792363 + 1.54342629115441*i
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