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La ecuación:
Ecuación (5x+8)-(8x+14)=9
Ecuación (x-1)/(5-x)=2/9
Ecuación 9-2*(-4*x+7)=7
Ecuación x^2-6x+9=0
Expresar {x} en función de y en la ecuación:
7*x+6*y=20
-14*x+5*y=-1
7*x-6*y=-2
3*x-7*y=13
Derivada de:
z^4
Gráfico de la función y =:
z^4
Expresiones idénticas
z^ cuatro =-3sqrt(3i)- nueve = cero
z en el grado 4 es igual a menos 3 raíz cuadrada de (3i) menos 9 es igual a 0
z en el grado cuatro es igual a menos 3 raíz cuadrada de (3i) menos nueve es igual a cero
z^4=-3√(3i)-9=0
z4=-3sqrt(3i)-9=0
z4=-3sqrt3i-9=0
z⁴=-3sqrt(3i)-9=0
z^4=-3sqrt3i-9=0
z^4=-3sqrt(3i)-9=O
Expresiones semejantes
z^4=+3sqrt(3i)-9=0
z^4=-3sqrt(3i)+9=0

z^4=-3sqrt(3i)-9=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

 4         _____    
z  = - 3*\/ 3*I  - 9

z^{4} = -9 - 3 \sqrt{3 i}

Simplificar

Solución detallada

Tenemos la ecuación

z^{4} = -9 - 3 \sqrt{3 i}

Ya que la potencia en la ecuación es igual a = 4 y miembro libre = -9 - 3*sqrt(3)*sqrt(i) complejo,
significa que la ecuación correspondiente no tiene soluciones reales

Las demás 4 raíces son complejas.
hacemos el cambio:

w = z

entonces la ecuación será así:

w^{4} = -9 - 3 \sqrt{3} \sqrt{i}

Cualquier número complejo se puede presentar que:

w = r e^{i p}

sustituimos en la ecuación

r^{4} e^{4 i p} = -9 - 3 \sqrt{3} \sqrt{i}

donde

r = \sqrt[8]{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}

- módulo del número complejo
Sustituyamos r:

e^{4 i p} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}

Usando la fórmula de Euler hallemos las raíces para p

i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = \frac{-9 - 3 \sqrt{3} \sqrt{i}}{\sqrt{\frac{27 \sqrt{6}}{2} + 81 - \frac{27 \sqrt{6} i}{2} - \frac{27 \sqrt{2} \sqrt{i} i}{2} + \frac{27 \sqrt{2} \sqrt{i}}{2} + 27 \sqrt{3} \sqrt{i}}}

es decir

\cos{\left(4 p \right)} = \frac{-9 - \frac{3 \sqrt{6}}{2}}{\sqrt{27 \sqrt{6} + 108}}

\sin{\left(4 p \right)} = - \frac{3 \sqrt{6}}{2 \sqrt{27 \sqrt{6} + 108}}

entonces

p = \frac{\pi N}{2} - \frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4}

donde N=0,1,2,3,...
Seleccionando los valores de N y sustituyendo p en la fórmula para w
Es decir, la solución será para w:

w_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

w_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

w_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

w_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

hacemos cambio inverso

w = z

z = w

Entonces la respuesta definitiva es:

z_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

z_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

z_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

z_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

Gráfica

Suma y producto de raíces [src]

suma

                         /         /        ___     \\                            /         /        ___     \\                          /         /        ___     \\                            /         /        ___     \\                            /         /        ___     \\                            /         /        ___     \\                          /         /        ___     \\                            /         /        ___     \\
                         |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||                          |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||                          |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||
                         |     atan|----------------||                            |     atan|----------------||                          |     atan|----------------||                            |     atan|----------------||                            |     atan|----------------||                            |     atan|----------------||                          |     atan|----------------||                            |     atan|----------------||
                         |         |  /         ___\||                            |         |  /         ___\||                          |         |  /         ___\||                            |         |  /         ___\||                            |         |  /         ___\||                            |         |  /         ___\||                          |         |  /         ___\||                            |         |  /         ___\||
                         |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||                          |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||                          |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||
     ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||      ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||      ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||
  8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|   8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|   8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|
- \/  108 + 27*\/ 6  *sin|-- + ----------------------| - I*\/  108 + 27*\/ 6  *cos|-- + ----------------------| + \/  108 + 27*\/ 6  *sin|-- + ----------------------| + I*\/  108 + 27*\/ 6  *cos|-- + ----------------------| + - \/  108 + 27*\/ 6  *cos|-- + ----------------------| + I*\/  108 + 27*\/ 6  *sin|-- + ----------------------| + \/  108 + 27*\/ 6  *cos|-- + ----------------------| - I*\/  108 + 27*\/ 6  *sin|-- + ----------------------|
                         \4              4           /                            \4              4           /                          \4              4           /                            \4              4           /                            \4              4           /                            \4              4           /                          \4              4           /                            \4              4           /

\left(\sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(\left(\left(- \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) + \left(\sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)\right) + \left(- \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)\right)

0

producto

/                         /         /        ___     \\                            /         /        ___     \\\ /                       /         /        ___     \\                            /         /        ___     \\\ /                         /         /        ___     \\                            /         /        ___     \\\ /                       /         /        ___     \\                            /         /        ___     \\\
|                         |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||| |                       |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||| |                         |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||| |                       |         |    3*\/ 6      ||                            |         |    3*\/ 6      |||
|                         |     atan|----------------||                            |     atan|----------------||| |                       |     atan|----------------||                            |     atan|----------------||| |                         |     atan|----------------||                            |     atan|----------------||| |                       |     atan|----------------||                            |     atan|----------------|||
|                         |         |  /         ___\||                            |         |  /         ___\||| |                       |         |  /         ___\||                            |         |  /         ___\||| |                         |         |  /         ___\||                            |         |  /         ___\||| |                       |         |  /         ___\||                            |         |  /         ___\|||
|                         |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||| |                       |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||| |                         |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||| |                       |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 ||||
|     ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||| |   ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||| |     ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||| |   ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------||||
|  8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|| |8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|| |  8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|| |8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //||
|- \/  108 + 27*\/ 6  *sin|-- + ----------------------| - I*\/  108 + 27*\/ 6  *cos|-- + ----------------------||*|\/  108 + 27*\/ 6  *sin|-- + ----------------------| + I*\/  108 + 27*\/ 6  *cos|-- + ----------------------||*|- \/  108 + 27*\/ 6  *cos|-- + ----------------------| + I*\/  108 + 27*\/ 6  *sin|-- + ----------------------||*|\/  108 + 27*\/ 6  *cos|-- + ----------------------| - I*\/  108 + 27*\/ 6  *sin|-- + ----------------------||
\                         \4              4           /                            \4              4           // \                       \4              4           /                            \4              4           // \                         \4              4           /                            \4              4           // \                       \4              4           /                            \4              4           //

\left(- \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(- \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right) \left(\sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}\right)

                                                                                    /           /      ___\\
                                                                             2      |           |1   \/ 6 ||
                    /     /         /      ___\\      /         /      ___\\\       |       atan|- - -----||
                    |     |         |1   \/ 6 ||      |         |1   \/ 6 |||       |  pi       \5     5  /|
     ______________ |     |     atan|- - -----||      |     atan|- - -----|||   2*I*|- -- - ---------------|
    /          ___  |     |pi       \5     5  /|      |pi       \5     5  /||       \  4           4       /
3*\/  12 + 3*\/ 6  *|I*cos|-- + ---------------| + sin|-- + ---------------|| *e                            
                    \     \4           4       /      \4           4       //

3 \sqrt{3 \sqrt{6} + 12} \left(\sin{\left(\frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4} + \frac{\pi}{4} \right)} + i \cos{\left(\frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4} + \frac{\pi}{4} \right)}\right)^{2} e^{2 i \left(- \frac{\pi}{4} - \frac{\operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{6}}{5} \right)}}{4}\right)}

3*sqrt(12 + 3*sqrt(6))*(i*cos(pi/4 + atan(1/5 - sqrt(6)/5)/4) + sin(pi/4 + atan(1/5 - sqrt(6)/5)/4))^2*exp(2*i*(-pi/4 - atan(1/5 - sqrt(6)/5)/4))

Respuesta rápida [src]

                              /         /        ___     \\                            /         /        ___     \\
                              |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||
                              |     atan|----------------||                            |     atan|----------------||
                              |         |  /         ___\||                            |         |  /         ___\||
                              |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||
          ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||
       8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|
z1 = - \/  108 + 27*\/ 6  *sin|-- + ----------------------| - I*\/  108 + 27*\/ 6  *cos|-- + ----------------------|
                              \4              4           /                            \4              4           /

z_{1} = - \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

                            /         /        ___     \\                            /         /        ___     \\
                            |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||
                            |     atan|----------------||                            |     atan|----------------||
                            |         |  /         ___\||                            |         |  /         ___\||
                            |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||
        ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||
     8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|
z2 = \/  108 + 27*\/ 6  *sin|-- + ----------------------| + I*\/  108 + 27*\/ 6  *cos|-- + ----------------------|
                            \4              4           /                            \4              4           /

z_{2} = \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

                              /         /        ___     \\                            /         /        ___     \\
                              |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||
                              |     atan|----------------||                            |     atan|----------------||
                              |         |  /         ___\||                            |         |  /         ___\||
                              |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||
          ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||
       8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|
z3 = - \/  108 + 27*\/ 6  *cos|-- + ----------------------| + I*\/  108 + 27*\/ 6  *sin|-- + ----------------------|
                              \4              4           /                            \4              4           /

z_{3} = - \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} + i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

                            /         /        ___     \\                            /         /        ___     \\
                            |         |    3*\/ 6      ||                            |         |    3*\/ 6      ||
                            |     atan|----------------||                            |     atan|----------------||
                            |         |  /         ___\||                            |         |  /         ___\||
                            |         |  |     3*\/ 6 |||                            |         |  |     3*\/ 6 |||
        ________________    |         |2*|-9 - -------|||        ________________    |         |2*|-9 - -------|||
     8 /            ___     |pi       \  \        2   //|     8 /            ___     |pi       \  \        2   //|
z4 = \/  108 + 27*\/ 6  *cos|-- + ----------------------| - I*\/  108 + 27*\/ 6  *sin|-- + ----------------------|
                            \4              4           /                            \4              4           /

z_{4} = \sqrt[8]{27 \sqrt{6} + 108} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)} - i \sqrt[8]{27 \sqrt{6} + 108} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{6}}{2 \left(-9 - \frac{3 \sqrt{6}}{2}\right)} \right)}}{4} + \frac{\pi}{4} \right)}

z4 = (27*sqrt(6) + 108)^(1/8)*cos(atan(3*sqrt(6)/(2*(-9 - 3*sqrt(6)/2)))/4 + pi/4) - i*(27*sqrt(6) + 108)^(1/8)*sin(atan(3*sqrt(6)/(2*(-9 - 3*sqrt(6)/2)))/4 + pi/4)

Respuesta numérica [src]

z1 = -1.24936764883673 - 1.43934700597303*i

z2 = 1.24936764883673 + 1.43934700597303*i

z3 = 1.43934700597303 - 1.24936764883673*i

z4 = -1.43934700597303 + 1.24936764883673*i

z4 = -1.43934700597303 + 1.24936764883673*i

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z^4=-3sqrt(3i)-9=0 la ecuación

Solución numérica:

Solución