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x^3+(3i*x^2)-(3*(1-2i)x)+10-5i=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x + 3*I*x - 3*(1 - 2*I)*x + 10 - 5*I = 0
$$\left(\left(- x 3 \left(1 - 2 i\right) + \left(x^{3} + 3 i x^{2}\right)\right) + 10\right) - 5 i = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 3 i$$
$$q = \frac{c}{a}$$
$$q = -3 + 6 i$$
$$v = \frac{d}{a}$$
$$v = 10 - 5 i$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - 3 i$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3 + 6 i$$
$$x_{1} x_{2} x_{3} = 10 - 5 i$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 3 i$$
$$q = \frac{c}{a}$$
$$q = -3 + 6 i$$
$$v = \frac{d}{a}$$
$$v = 10 - 5 i$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = - 3 i$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3 + 6 i$$
$$x_{1} x_{2} x_{3} = 10 - 5 i$$
Gráfica
Respuesta rápida
[src]
/ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\\ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\
| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||
| | | \ \ 2 // || | | \ \ 2 // ||| | | \ \ 2 // || | | \ \ 2 // ||
| |atan|----------------------------------------|| |atan|----------------------------------------||| |atan|----------------------------------------|| |atan|----------------------------------------||
| ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||| ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||
| / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||| / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||
| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /| ___ / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /| ___ / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|
| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *sin|----------------------------------------------| \/ 3 *6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *cos|----------------------------------------------|| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *cos|----------------------------------------------| \/ 3 *6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *sin|----------------------------------------------|
/ 1 \ | / 1 \ \/ \ \ 2 // \ \ 2 // \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /| \/ \ \ 2 // \ \ 2 // \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /
x1 = - 6*im|-------------------------------------------------------------------------------------------------------| + I*|-1 + 6*re|-------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------
| _________________________________________________________________________________| | | _________________________________________________________________________________| 6 6 | 6 6
| / ____________________________________________ | | | / ____________________________________________ | |
|/ ___\ / / 2 | | |/ ___\ / / 2 | |
|| 1 I*\/ 3 | / \/ (270 - 189*I - 27*I*(-3 + 6*I)) - 23328*I 189*I 27*I*(-3 + 6*I) | | || 1 I*\/ 3 | / \/ (270 - 189*I - 27*I*(-3 + 6*I)) - 23328*I 189*I 27*I*(-3 + 6*I) | |
||- - - -------|*3 / 135 + ----------------------------------------------- - ----- - --------------- | | ||- - - -------|*3 / 135 + ----------------------------------------------- - ----- - --------------- | |
\\ 2 2 / \/ 2 2 2 / \ \\ 2 2 / \/ 2 2 2 / /
$$x_{1} = - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{135 - \frac{189 i}{2} + \frac{\sqrt{\left(270 - 189 i - 27 i \left(-3 + 6 i\right)\right)^{2} - 23328 i}}{2} - \frac{27 i \left(-3 + 6 i\right)}{2}}}\right)} + \frac{\sqrt{3} \sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} + \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} + i \left(-1 + 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{135 - \frac{189 i}{2} + \frac{\sqrt{\left(270 - 189 i - 27 i \left(-3 + 6 i\right)\right)^{2} - 23328 i}}{2} - \frac{27 i \left(-3 + 6 i\right)}{2}}}\right)} - \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} + \frac{\sqrt{3} \sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6}\right)$$
/ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\\ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\
| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||
| | | \ \ 2 // || | | \ \ 2 // ||| | | \ \ 2 // || | | \ \ 2 // ||
| |atan|----------------------------------------|| |atan|----------------------------------------||| |atan|----------------------------------------|| |atan|----------------------------------------||
| | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||| | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||
| | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||| | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||
| | \ \ 2 / /| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|| | \ \ 2 / /| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|
| 6*cos|----------------------------------------------| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *sin|----------------------------------------------|| 6*sin|----------------------------------------------| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *cos|----------------------------------------------|
| \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /| \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /
x2 = I*|-1 + ---------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------| - ---------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------
| ___________________________________________________________________________________ 3 | ___________________________________________________________________________________ 3
| / 2 2 | / 2 2
| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\
| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| | 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------||
\ \/ \ \ 2 // \ \ 2 // / \/ \ \ 2 // \ \ 2 //
$$x_{2} = - \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{3} - \frac{6 \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}}} + i \left(-1 + \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{3} + \frac{6 \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}}}\right)$$
/ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\\ / / / ___ 4 ____ /atan(2/3)\\ \\ / / / ___ 4 ____ /atan(2/3)\\ \\
| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||| | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| || | |-|-54 - 54*\/ 5 *\/ 13 *sin|---------|| ||
| | | \ \ 2 // || | | \ \ 2 // ||| | | \ \ 2 // || | | \ \ 2 // ||
| |atan|----------------------------------------|| |atan|----------------------------------------||| |atan|----------------------------------------|| |atan|----------------------------------------||
| ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||| ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ || ___________________________________________________________________________________ | | ___ 4 ____ /atan(2/3)\ ||
| / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||| / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| || / 2 2 | | 216 + 54*\/ 5 *\/ 13 *cos|---------| ||
| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /| ___ / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|| / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /| ___ / / ___ 4 ____ /atan(2/3)\\ / ___ 4 ____ /atan(2/3)\\ | \ \ 2 / /|
| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *sin|----------------------------------------------| \/ 3 *6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *cos|----------------------------------------------|| 6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *cos|----------------------------------------------| \/ 3 *6 / |-54 - 54*\/ 5 *\/ 13 *sin|---------|| + |216 + 54*\/ 5 *\/ 13 *cos|---------|| *sin|----------------------------------------------|
/ 1 \ | / 1 \ \/ \ \ 2 // \ \ 2 // \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /| \/ \ \ 2 // \ \ 2 // \ 3 / \/ \ \ 2 // \ \ 2 // \ 3 /
x3 = - 6*im|-------------------------------------------------------------------------------------------------------| + I*|-1 + 6*re|-------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------
| _________________________________________________________________________________| | | _________________________________________________________________________________| 6 6 | 6 6
| / ____________________________________________ | | | / ____________________________________________ | |
|/ ___\ / / 2 | | |/ ___\ / / 2 | |
|| 1 I*\/ 3 | / \/ (270 - 189*I - 27*I*(-3 + 6*I)) - 23328*I 189*I 27*I*(-3 + 6*I) | | || 1 I*\/ 3 | / \/ (270 - 189*I - 27*I*(-3 + 6*I)) - 23328*I 189*I 27*I*(-3 + 6*I) | |
||- - + -------|*3 / 135 + ----------------------------------------------- - ----- - --------------- | | ||- - + -------|*3 / 135 + ----------------------------------------------- - ----- - --------------- | |
\\ 2 2 / \/ 2 2 2 / \ \\ 2 2 / \/ 2 2 2 / /
$$x_{3} = - \frac{\sqrt{3} \sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} - 6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{135 - \frac{189 i}{2} + \frac{\sqrt{\left(270 - 189 i - 27 i \left(-3 + 6 i\right)\right)^{2} - 23328 i}}{2} - \frac{27 i \left(-3 + 6 i\right)}{2}}}\right)} + \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} + i \left(- \frac{\sqrt{3} \sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6} - 1 + 6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{135 - \frac{189 i}{2} + \frac{\sqrt{\left(270 - 189 i - 27 i \left(-3 + 6 i\right)\right)^{2} - 23328 i}}{2} - \frac{27 i \left(-3 + 6 i\right)}{2}}}\right)} - \frac{\sqrt[6]{\left(- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54\right)^{2} + \left(216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan}{\left(- \frac{- 54 \sqrt[4]{13} \sqrt{5} \sin{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)} - 54}{216 + 54 \sqrt[4]{13} \sqrt{5} \cos{\left(\frac{\operatorname{atan}{\left(\frac{2}{3} \right)}}{2} \right)}} \right)}}{3} \right)}}{6}\right)$$
x3 = -sqrt(3)*((-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)^2 + (216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2))^2)^(1/6)*sin(atan(-(-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)/(216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2)))/3)/6 - 6*im(1/((-1/2 + sqrt(3)*i/2)*(135 - 189*i/2 + sqrt((270 - 189*i - 27*i*(-3 + 6*i))^2 - 23328*i)/2 - 27*i*(-3 + 6*i)/2)^(1/3))) + ((-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)^2 + (216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2))^2)^(1/6)*cos(atan(-(-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)/(216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2)))/3)/6 + i*(-sqrt(3)*((-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)^2 + (216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2))^2)^(1/6)*cos(atan(-(-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)/(216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2)))/3)/6 - 1 + 6*re(1/((-1/2 + sqrt(3)*i/2)*(135 - 189*i/2 + sqrt((270 - 189*i - 27*i*(-3 + 6*i))^2 - 23328*i)/2 - 27*i*(-3 + 6*i)/2)^(1/3))) - ((-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)^2 + (216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2))^2)^(1/6)*sin(atan(-(-54*13^(1/4)*sqrt(5)*sin(atan(2/3)/2) - 54)/(216 + 54*13^(1/4)*sqrt(5)*cos(atan(2/3)/2)))/3)/6)
Respuesta numérica
[src]
x1 = 1.78428740225808 - 3.64955865171258*i
x2 = 0.833361970306632 + 0.641023196786008*i
x3 = -2.61764937256471 + 0.00853545492657097*i
x3 = -2.61764937256471 + 0.00853545492657097*i