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x^3-2022x+sqrt(2021)=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Solución numérica:

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Solución

Ha introducido [src] 
 3              ______    
x  - 2022*x + \/ 2021  = 0
(x3−2022x)+2021=0\left(x^{3} - 2022 x\right) + \sqrt{2021} = 0
Simplificar
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=−2022q = -2022
v=dav = \frac{d}{a}
v=2021v = \sqrt{2021}
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=−2022x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2022
x1x2x3=2021x_{1} x_{2} x_{3} = \sqrt{2021}
Suma y producto de raíces [src] 
suma
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                                                                   |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||                                                                      |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||                                                ______________________________
                                                                   |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||                                                                      |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||                                               /      ______              ___ 
                                                                   |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|                                                                      |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|                                              /  27*\/ 2021    545535*I*\/ 3  
                                                                   |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|                                                                      |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|                                           3 /   ----------- + -------------- 
         /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /            /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /                     2022                  \/         2              2        
- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ---------------------------------- + - 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ---------------------------------- + - ----------------------------------- - -----------------------------------
         |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                             |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                           ______________________________                    3                 
         |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                                     |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                                  /      ______              ___                                       
         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                                     ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                                 /  27*\/ 2021    545535*I*\/ 3                                        
         ||- - - -------|*3 /   ----------- + -------------- |     |         ||- - - -------|*3 /   ----------- + -------------- |                                                                         |                                                                                     ||- - + -------|*3 /   ----------- + -------------- |     |         ||- - + -------|*3 /   ----------- + -------------- |                                                                         |                                                                              3 /   ----------- + --------------                                       
         \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /                                                                                     \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /                                                                              \/         2              2
((−2022sin⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12−3i2)2720212+5455353i23)+674cos⁡(atan⁡(2020560632021)3)2+i(−2022im⁡(1(−12−3i2)2720212+5455353i23)+674sin⁡(atan⁡(2020560632021)3)2+2022cos⁡(atan⁡(2020560632021)3)2))+(2022sin⁡(atan⁡(2020560632021)3)2+674cos⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12+3i2)2720212+5455353i23)+i(−2022cos⁡(atan⁡(2020560632021)3)2+674sin⁡(atan⁡(2020560632021)3)2−2022im⁡(1(−12+3i2)2720212+5455353i23))))+(−2720212+5455353i233−20222720212+5455353i23)\left(\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) + \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right) 
=
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                                                                                                                                                                           ______________________________     |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||     |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||                                    
                                                                                                                                                                          /      ______              ___      |                                                                            |atan|--------------||               |atan|--------------|||     |                                                                            |atan|--------------||               |atan|--------------|||              /    /        ______\\
                                                                                                                                                                         /  27*\/ 2021    545535*I*\/ 3       |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||              |    |20205*\/ 6063 ||
                                                                                                                                                                      3 /   ----------- + --------------      |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||     |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||              |atan|--------------||
                  2022                         /                         1                         \          /                         1                         \   \/         2              2             |         /                         1                         \              \         3          /               \         3          /|     |         /                         1                         \              \         3          /               \         3          /|     _____    |    \     2021     /|
- ----------------------------------- - 2022*re|---------------------------------------------------| - 2022*re|---------------------------------------------------| - ----------------------------------- + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + \/ 674 *cos|--------------------|
       ______________________________          |                     ______________________________|          |                     ______________________________|                    3                      |         |                     ______________________________|                   2                                   2                 |     |         |                     ______________________________|                   2                                   2                 |              \         3          /
      /      ______              ___           |/          ___\     /      ______              ___ |          |/          ___\     /      ______              ___ |                                           |         |/          ___\     /      ______              ___ |                                                                         |     |         |/          ___\     /      ______              ___ |                                                                         |                                    
     /  27*\/ 2021    545535*I*\/ 3            ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |          ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                           |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                    
  3 /   ----------- + --------------           ||- - + -------|*3 /   ----------- + -------------- |          ||- - - -------|*3 /   ----------- + -------------- |                                           |         ||- - + -------|*3 /   ----------- + -------------- |                                                                         |     |         ||- - - -------|*3 /   ----------- + -------------- |                                                                         |                                    
  \/         2              2                  \\  2      2   / \/         2              2        /          \\  2      2   / \/         2              2        /                                           \         \\  2      2   / \/         2              2        /                                                                         /     \         \\  2      2   / \/         2              2        /                                                                         /
−2022re⁡(1(−12−3i2)2720212+5455353i23)−2022re⁡(1(−12+3i2)2720212+5455353i23)+674cos⁡(atan⁡(2020560632021)3)−2720212+5455353i233+i(−2022im⁡(1(−12−3i2)2720212+5455353i23)+674sin⁡(atan⁡(2020560632021)3)2+2022cos⁡(atan⁡(2020560632021)3)2)+i(−2022cos⁡(atan⁡(2020560632021)3)2+674sin⁡(atan⁡(2020560632021)3)2−2022im⁡(1(−12+3i2)2720212+5455353i23))−20222720212+5455353i23- 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)} - \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right) - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}} 
producto
/                                                                   /                                                                            /    /        ______\\               /    /        ______\\\              /    /        ______\\               /    /        ______\\\ /                                                                   /                                                                            /    /        ______\\               /    /        ______\\\              /    /        ______\\               /    /        ______\\\                                                                              
|                                                                   |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||| |                                                                   |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||| /                                             ______________________________\
|                                                                   |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||| |                                                                   |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||| |                                            /      ______              ___ |
|                                                                   |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|| |                                                                   |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|| |                                           /  27*\/ 2021    545535*I*\/ 3  |
|                                                                   |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|| |                                                                   |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|| |                                        3 /   ----------- + -------------- |
|         /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /| |         /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /| |                  2022                  \/         2              2        |
|- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ----------------------------------|*|- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ----------------------------------|*|- ----------------------------------- - -----------------------------------|
|         |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                 | |         |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                 | |       ______________________________                    3                 |
|         |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                         | |         |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                         | |      /      ______              ___                                       |
|         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                         | |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                         | |     /  27*\/ 2021    545535*I*\/ 3                                        |
|         ||- - - -------|*3 /   ----------- + -------------- |     |         ||- - - -------|*3 /   ----------- + -------------- |                                                                         |                                                                         | |         ||- - + -------|*3 /   ----------- + -------------- |     |         ||- - + -------|*3 /   ----------- + -------------- |                                                                         |                                                                         | |  3 /   ----------- + --------------                                       |
\         \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /                                                                         / \         \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /                                                                         / \  \/         2              2                                              /
(−2022sin⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12−3i2)2720212+5455353i23)+674cos⁡(atan⁡(2020560632021)3)2+i(−2022im⁡(1(−12−3i2)2720212+5455353i23)+674sin⁡(atan⁡(2020560632021)3)2+2022cos⁡(atan⁡(2020560632021)3)2))(2022sin⁡(atan⁡(2020560632021)3)2+674cos⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12+3i2)2720212+5455353i23)+i(−2022cos⁡(atan⁡(2020560632021)3)2+674sin⁡(atan⁡(2020560632021)3)2−2022im⁡(1(−12+3i2)2720212+5455353i23)))(−2720212+5455353i233−20222720212+5455353i23)\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right) 
=
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          |         |      |20205*\/ 6063 |||                                             
          |         |2*atan|--------------||| /                                       2/3\
    3 ___ |         |      \     2021     /|| |       3 ___ /  ______             ___\   |
337*\/ 2 *|1 - 2*cos|----------------------||*\1348 + \/ 2 *\\/ 2021  + 20205*I*\/ 3 /   /
          \         \          3           //                                             
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                              3 /   ______             ___                                
                              \/  \/ 2021  + 20205*I*\/ 3
33723(1−2cos⁡(2atan⁡(2020560632021)3))(1348+23(2021+202053i)23)2021+202053i3\frac{337 \sqrt[3]{2} \left(1 - 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}\right) \left(1348 + \sqrt[3]{2} \left(\sqrt{2021} + 20205 \sqrt{3} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{\sqrt{2021} + 20205 \sqrt{3} i}} 
337*2^(1/3)*(1 - 2*cos(2*atan(20205*sqrt(6063)/2021)/3))*(1348 + 2^(1/3)*(sqrt(2021) + 20205*i*sqrt(3))^(2/3))/(sqrt(2021) + 20205*i*sqrt(3))^(1/3)
Respuesta rápida [src] 
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                                                                        |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||
                                                                        |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||
                                                                        |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|
                                                                        |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|
              /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /
x1 = - 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ----------------------------------
              |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                 
              |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                         
              ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                         
              ||- - - -------|*3 /   ----------- + -------------- |     |         ||- - - -------|*3 /   ----------- + -------------- |                                                                         |                                                                         
              \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /
x1=−2022sin⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12−3i2)2720212+5455353i23)+674cos⁡(atan⁡(2020560632021)3)2+i(−2022im⁡(1(−12−3i2)2720212+5455353i23)+674sin⁡(atan⁡(2020560632021)3)2+2022cos⁡(atan⁡(2020560632021)3)2)x_{1} = - \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right) 
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                                                                        |                                                                            |    |20205*\/ 6063 ||               |    |20205*\/ 6063 |||              |    |20205*\/ 6063 ||               |    |20205*\/ 6063 ||
                                                                        |                                                                            |atan|--------------||               |atan|--------------|||              |atan|--------------||               |atan|--------------||
                                                                        |                                                                   _____    |    \     2021     /|     ______    |    \     2021     /||     _____    |    \     2021     /|     ______    |    \     2021     /|
                                                                        |                                                                 \/ 674 *sin|--------------------|   \/ 2022 *cos|--------------------||   \/ 674 *cos|--------------------|   \/ 2022 *sin|--------------------|
              /                         1                         \     |         /                         1                         \              \         3          /               \         3          /|              \         3          /               \         3          /
x2 = - 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ----------------------------------
              |                     ______________________________|     |         |                     ______________________________|                   2                                   2                 |                   2                                   2                 
              |/          ___\     /      ______              ___ |     |         |/          ___\     /      ______              ___ |                                                                         |                                                                         
              ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |     |         ||  1   I*\/ 3 |    /  27*\/ 2021    545535*I*\/ 3  |                                                                         |                                                                         
              ||- - + -------|*3 /   ----------- + -------------- |     |         ||- - + -------|*3 /   ----------- + -------------- |                                                                         |                                                                         
              \\  2      2   / \/         2              2        /     \         \\  2      2   / \/         2              2        /                                                                         /
x2=2022sin⁡(atan⁡(2020560632021)3)2+674cos⁡(atan⁡(2020560632021)3)2−2022re⁡(1(−12+3i2)2720212+5455353i23)+i(−2022cos⁡(atan⁡(2020560632021)3)2+674sin⁡(atan⁡(2020560632021)3)2−2022im⁡(1(−12+3i2)2720212+5455353i23))x_{2} = \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right) 
                                                  ______________________________
                                                 /      ______              ___ 
                                                /  27*\/ 2021    545535*I*\/ 3  
                                             3 /   ----------- + -------------- 
                       2022                  \/         2              2        
x3 = - ----------------------------------- - -----------------------------------
            ______________________________                    3                 
           /      ______              ___                                       
          /  27*\/ 2021    545535*I*\/ 3                                        
       3 /   ----------- + --------------                                       
       \/         2              2
x3=−2720212+5455353i233−20222720212+5455353i23x_{3} = - \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}} 
x3 = -(27*sqrt(2021)/2 + 545535*sqrt(3)*i/2)^(1/3)/3 - 2022/(27*sqrt(2021)/2 + 545535*sqrt(3)*i/2)^(1/3)
Respuesta numérica [src] 
x1 = 44.955533585978 + 1.0842021724855e-19*i
x2 = -44.977766792989
x3 = 0.0222332070109776
x3 = 0.0222332070109776
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