x^3-x+3=0 (уравнение)

Найду корень уравнения: x^3-x+3=0

Вы ввели

 3            
x  - x + 3 = 0

$$x^{3} - x + 3 = 0$$

Быстрый ответ

                                    ________________     /                                       ________________\
                                   /          _____      |                                      /          _____ |
                                  /  81   3*\/ 717       |                              ___    /  81   3*\/ 717  |
                               3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |
                1              \/    2        2          |           \/ 3                   \/    2        2     |
x1 = ----------------------- + --------------------- + I*|- ----------------------- + ---------------------------|
            ________________             6               |         ________________                6             |
           /          _____                              |        /          _____                               |
          /  81   3*\/ 717                               |       /  81   3*\/ 717                                |
     2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               |
       \/    2        2                                  \    \/    2        2                                   /

$$x_{1} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)$$

                                    ________________     /                                     ________________\
                                   /          _____      |                                    /          _____ |
                                  /  81   3*\/ 717       |                            ___    /  81   3*\/ 717  |
                               3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |
                1              \/    2        2          |         \/ 3                   \/    2        2     |
x2 = ----------------------- + --------------------- + I*|----------------------- - ---------------------------|
            ________________             6               |       ________________                6             |
           /          _____                              |      /          _____                               |
          /  81   3*\/ 717                               |     /  81   3*\/ 717                                |
     2*3 /   -- + ---------                              |2*3 /   -- + ---------                               |
       \/    2        2                                  \  \/    2        2                                   /

$$x_{2} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)$$

                                    ________________
                                   /          _____ 
                                  /  81   3*\/ 717  
                               3 /   -- + --------- 
                 1             \/    2        2     
x3 = - --------------------- - ---------------------
            ________________             3          
           /          _____                         
          /  81   3*\/ 717                          
       3 /   -- + ---------                         
       \/    2        2                             

$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}$$

Сумма и произведение корней

сумма

                                   ________________     /                                       ________________\                                  ________________     /                                     ________________\                                  ________________
                                  /          _____      |                                      /          _____ |                                 /          _____      |                                    /          _____ |                                 /          _____ 
                                 /  81   3*\/ 717       |                              ___    /  81   3*\/ 717  |                                /  81   3*\/ 717       |                            ___    /  81   3*\/ 717  |                                /  81   3*\/ 717  
                              3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |                             3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |                             3 /   -- + --------- 
               1              \/    2        2          |           \/ 3                   \/    2        2     |              1              \/    2        2          |         \/ 3                   \/    2        2     |               1             \/    2        2     
0 + ----------------------- + --------------------- + I*|- ----------------------- + ---------------------------| + ----------------------- + --------------------- + I*|----------------------- - ---------------------------| + - --------------------- - ---------------------
           ________________             6               |         ________________                6             |          ________________             6               |       ________________                6             |          ________________             3          
          /          _____                              |        /          _____                               |         /          _____                              |      /          _____                               |         /          _____                         
         /  81   3*\/ 717                               |       /  81   3*\/ 717                                |        /  81   3*\/ 717                               |     /  81   3*\/ 717                                |        /  81   3*\/ 717                          
    2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               |   2*3 /   -- + ---------                              |2*3 /   -- + ---------                               |     3 /   -- + ---------                         
      \/    2        2                                  \    \/    2        2                                   /     \/    2        2                                  \  \/    2        2                                   /     \/    2        2                             

$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right) + \left(\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)\right) + \left(0 + \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)\right)\right)\right)$$

=

  /                                     ________________\     /                                       ________________\
  |                                    /          _____ |     |                                      /          _____ |
  |                            ___    /  81   3*\/ 717  |     |                              ___    /  81   3*\/ 717  |
  |           ___            \/ 3 *3 /   -- + --------- |     |             ___            \/ 3 *3 /   -- + --------- |
  |         \/ 3                   \/    2        2     |     |           \/ 3                   \/    2        2     |
I*|----------------------- - ---------------------------| + I*|- ----------------------- + ---------------------------|
  |       ________________                6             |     |         ________________                6             |
  |      /          _____                               |     |        /          _____                               |
  |     /  81   3*\/ 717                                |     |       /  81   3*\/ 717                                |
  |2*3 /   -- + ---------                               |     |  2*3 /   -- + ---------                               |
  \  \/    2        2                                   /     \    \/    2        2                                   /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)$$

произведение

  /                               ________________     /                                       ________________\\ /                               ________________     /                                     ________________\\ /                               ________________\
  |                              /          _____      |                                      /          _____ || |                              /          _____      |                                    /          _____ || |                              /          _____ |
  |                             /  81   3*\/ 717       |                              ___    /  81   3*\/ 717  || |                             /  81   3*\/ 717       |                            ___    /  81   3*\/ 717  || |                             /  81   3*\/ 717  |
  |                          3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- || |                          3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- || |                          3 /   -- + --------- |
  |           1              \/    2        2          |           \/ 3                   \/    2        2     || |           1              \/    2        2          |         \/ 3                   \/    2        2     || |            1             \/    2        2     |
1*|----------------------- + --------------------- + I*|- ----------------------- + ---------------------------||*|----------------------- + --------------------- + I*|----------------------- - ---------------------------||*|- --------------------- - ---------------------|
  |       ________________             6               |         ________________                6             || |       ________________             6               |       ________________                6             || |       ________________             3          |
  |      /          _____                              |        /          _____                               || |      /          _____                              |      /          _____                               || |      /          _____                         |
  |     /  81   3*\/ 717                               |       /  81   3*\/ 717                                || |     /  81   3*\/ 717                               |     /  81   3*\/ 717                                || |     /  81   3*\/ 717                          |
  |2*3 /   -- + ---------                              |  2*3 /   -- + ---------                               || |2*3 /   -- + ---------                              |2*3 /   -- + ---------                               || |  3 /   -- + ---------                         |
  \  \/    2        2                                  \    \/    2        2                                   // \  \/    2        2                                  \  \/    2        2                                   // \  \/    2        2                             /

$$1 \cdot \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6}\right)\right) \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{717}}{2} + \frac{81}{2}}}\right)$$

=

           __________________              ________________                          2/3                __________________                           2/3                                 2/3                     ________________                                     2/3
        3 /            _____        2/3 3 /          _____      3 ____ /       _____\          _____ 3 /            _____      3 ___ /         _____\      3 ___   _____ /         _____\         2/3   _____ 3 /          _____    3 ___ 6 ___   _____ /       _____\   
     81*\/  324 + 12*\/ 717     81*2   *\/  81 + 3*\/ 717     3*\/ 18 *\27 + \/ 717 /      3*\/ 717 *\/  324 + 12*\/ 717     3*\/ 2 *\81 + 3*\/ 717 /      \/ 2 *\/ 717 *\81 + 3*\/ 717 /      3*2   *\/ 717 *\/  81 + 3*\/ 717     \/ 2 *\/ 3 *\/ 239 *\27 + \/ 717 /   
-3 + ------------------------ - --------------------------- - -------------------------- - ------------------------------- + --------------------------- - --------------------------------- + ---------------------------------- + -------------------------------------
                32                           32                           8                               32                              8                                72                                  32                                     24                 

$$- \frac{81 \cdot 2^{\frac{2}{3}} \sqrt[3]{3 \sqrt{717} + 81}}{32} - \frac{3 \sqrt{717} \sqrt[3]{12 \sqrt{717} + 324}}{32} - \frac{3 \cdot \sqrt[3]{18} \left(\sqrt{717} + 27\right)^{\frac{2}{3}}}{8} - \frac{\sqrt[3]{2} \sqrt{717} \left(3 \sqrt{717} + 81\right)^{\frac{2}{3}}}{72} - 3 + \frac{\sqrt[3]{2} \sqrt{239} \cdot \sqrt[6]{3} \left(\sqrt{717} + 27\right)^{\frac{2}{3}}}{24} + \frac{3 \cdot \sqrt[3]{2} \left(3 \sqrt{717} + 81\right)^{\frac{2}{3}}}{8} + \frac{3 \cdot 2^{\frac{2}{3}} \sqrt{717} \sqrt[3]{3 \sqrt{717} + 81}}{32} + \frac{81 \sqrt[3]{12 \sqrt{717} + 324}}{32}$$

Теорема Виета

это приведённое кубическое уравнение
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = 3$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = 3$$

Численный ответ

x1 = 0.83584994082858 - 1.04686931884998*i

x2 = -1.67169988165716

x3 = 0.83584994082858 + 1.04686931884998*i

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