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

            _____________                            /             _____________                       \
           /       _____                             |            /       _____                        |
          /  3   \/ 717                              |    ___    /  3   \/ 717                         |
       3 /   - + -------                             |  \/ 3 *3 /   - + -------             ___        |
       \/    2      18               1               |        \/    2      18             \/ 3         |
x1 = - ------------------ - -------------------- + I*|- ------------------------ + --------------------|
               2                   _____________     |             2                      _____________|
                                  /       _____      |                                   /       _____ |
                                 /  3   \/ 717       |                                  /  3   \/ 717  |
                            6*3 /   - + -------      |                             6*3 /   - + ------- |
                              \/    2      18        \                               \/    2      18   /

$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)$$

            _____________                            /           _____________                       \
           /       _____                             |          /       _____                        |
          /  3   \/ 717                              |  ___    /  3   \/ 717                         |
       3 /   - + -------                             |\/ 3 *3 /   - + -------             ___        |
       \/    2      18               1               |      \/    2      18             \/ 3         |
x2 = - ------------------ - -------------------- + I*|------------------------ - --------------------|
               2                   _____________     |           2                      _____________|
                                  /       _____      |                                 /       _____ |
                                 /  3   \/ 717       |                                /  3   \/ 717  |
                            6*3 /   - + -------      |                           6*3 /   - + ------- |
                              \/    2      18        \                             \/    2      18   /

$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)$$

          _____________                       
         /       _____                        
        /  3   \/ 717              1          
x3 = 3 /   - + -------  + --------------------
     \/    2      18             _____________
                                /       _____ 
                               /  3   \/ 717  
                          3*3 /   - + ------- 
                            \/    2      18   

$$x_{3} = \frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}$$

Сумма и произведение корней

сумма

           _____________                            /             _____________                       \          _____________                            /           _____________                       \                                            
          /       _____                             |            /       _____                        |         /       _____                             |          /       _____                        |                                            
         /  3   \/ 717                              |    ___    /  3   \/ 717                         |        /  3   \/ 717                              |  ___    /  3   \/ 717                         |        _____________                       
      3 /   - + -------                             |  \/ 3 *3 /   - + -------             ___        |     3 /   - + -------                             |\/ 3 *3 /   - + -------             ___        |       /       _____                        
      \/    2      18               1               |        \/    2      18             \/ 3         |     \/    2      18               1               |      \/    2      18             \/ 3         |      /  3   \/ 717              1          
0 + - ------------------ - -------------------- + I*|- ------------------------ + --------------------| + - ------------------ - -------------------- + I*|------------------------ - --------------------| + 3 /   - + -------  + --------------------
              2                   _____________     |             2                      _____________|             2                   _____________     |           2                      _____________|   \/    2      18             _____________
                                 /       _____      |                                   /       _____ |                                /       _____      |                                 /       _____ |                              /       _____ 
                                /  3   \/ 717       |                                  /  3   \/ 717  |                               /  3   \/ 717       |                                /  3   \/ 717  |                             /  3   \/ 717  
                           6*3 /   - + -------      |                             6*3 /   - + ------- |                          6*3 /   - + -------      |                           6*3 /   - + ------- |                        3*3 /   - + ------- 
                             \/    2      18        \                               \/    2      18   /                            \/    2      18        \                             \/    2      18   /                          \/    2      18   

$$\left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}\right) - \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}} - i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right) - i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)\right)$$

=

  /           _____________                       \     /             _____________                       \
  |          /       _____                        |     |            /       _____                        |
  |  ___    /  3   \/ 717                         |     |    ___    /  3   \/ 717                         |
  |\/ 3 *3 /   - + -------             ___        |     |  \/ 3 *3 /   - + -------             ___        |
  |      \/    2      18             \/ 3         |     |        \/    2      18             \/ 3         |
I*|------------------------ - --------------------| + I*|- ------------------------ + --------------------|
  |           2                      _____________|     |             2                      _____________|
  |                                 /       _____ |     |                                   /       _____ |
  |                                /  3   \/ 717  |     |                                  /  3   \/ 717  |
  |                           6*3 /   - + ------- |     |                             6*3 /   - + ------- |
  \                             \/    2      18   /     \                               \/    2      18   /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right) + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)$$

произведение

  /       _____________                            /             _____________                       \\ /       _____________                            /           _____________                       \\                                            
  |      /       _____                             |            /       _____                        || |      /       _____                             |          /       _____                        ||                                            
  |     /  3   \/ 717                              |    ___    /  3   \/ 717                         || |     /  3   \/ 717                              |  ___    /  3   \/ 717                         || /     _____________                       \
  |  3 /   - + -------                             |  \/ 3 *3 /   - + -------             ___        || |  3 /   - + -------                             |\/ 3 *3 /   - + -------             ___        || |    /       _____                        |
  |  \/    2      18               1               |        \/    2      18             \/ 3         || |  \/    2      18               1               |      \/    2      18             \/ 3         || |   /  3   \/ 717              1          |
1*|- ------------------ - -------------------- + I*|- ------------------------ + --------------------||*|- ------------------ - -------------------- + I*|------------------------ - --------------------||*|3 /   - + -------  + --------------------|
  |          2                   _____________     |             2                      _____________|| |          2                   _____________     |           2                      _____________|| |\/    2      18             _____________|
  |                             /       _____      |                                   /       _____ || |                             /       _____      |                                 /       _____ || |                           /       _____ |
  |                            /  3   \/ 717       |                                  /  3   \/ 717  || |                            /  3   \/ 717       |                                /  3   \/ 717  || |                          /  3   \/ 717  |
  |                       6*3 /   - + -------      |                             6*3 /   - + ------- || |                       6*3 /   - + -------      |                           6*3 /   - + ------- || |                     3*3 /   - + ------- |
  \                         \/    2      18        \                               \/    2      18   // \                         \/    2      18        \                             \/    2      18   // \                       \/    2      18   /

$$\left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2}\right)\right) 1 \left(- \frac{\sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}}\right)\right) \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}} + \sqrt[3]{\frac{\sqrt{717}}{18} + \frac{3}{2}}\right)$$

=

3

$$3$$

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

График