x^3+x-1=0 (уравнение)

Найду корень уравнения: x^3+x-1=0

Вы ввели

 3            
x  + x - 1 = 0

$$x^{3} + x - 1 = 0$$

Быстрый ответ

            ____________                           /             ____________                      \
           /       ____                            |            /       ____                       |
          /  1   \/ 93                             |    ___    /  1   \/ 93                        |
       3 /   - + ------                            |  \/ 3 *3 /   - + ------             ___       |
       \/    2     18               1              |        \/    2     18             \/ 3        |
x1 = - ----------------- + ------------------- + I*|- ----------------------- - -------------------|
               2                  ____________     |             2                     ____________|
                                 /       ____      |                                  /       ____ |
                                /  1   \/ 93       |                                 /  1   \/ 93  |
                           6*3 /   - + ------      |                            6*3 /   - + ------ |
                             \/    2     18        \                              \/    2     18   /

$$x_{1} = - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)$$

            ____________                           /           ____________                      \
           /       ____                            |          /       ____                       |
          /  1   \/ 93                             |  ___    /  1   \/ 93                        |
       3 /   - + ------                            |\/ 3 *3 /   - + ------             ___       |
       \/    2     18               1              |      \/    2     18             \/ 3        |
x2 = - ----------------- + ------------------- + I*|----------------------- + -------------------|
               2                  ____________     |           2                     ____________|
                                 /       ____      |                                /       ____ |
                                /  1   \/ 93       |                               /  1   \/ 93  |
                           6*3 /   - + ------      |                          6*3 /   - + ------ |
                             \/    2     18        \                            \/    2     18   /

$$x_{2} = - \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)$$

          ____________                      
         /       ____                       
        /  1   \/ 93              1         
x3 = 3 /   - + ------  - -------------------
     \/    2     18             ____________
                               /       ____ 
                              /  1   \/ 93  
                         3*3 /   - + ------ 
                           \/    2     18   

$$x_{3} = - \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}$$

Сумма и произведение корней

сумма

           ____________                           /             ____________                      \          ____________                           /           ____________                      \                                          
          /       ____                            |            /       ____                       |         /       ____                            |          /       ____                       |                                          
         /  1   \/ 93                             |    ___    /  1   \/ 93                        |        /  1   \/ 93                             |  ___    /  1   \/ 93                        |        ____________                      
      3 /   - + ------                            |  \/ 3 *3 /   - + ------             ___       |     3 /   - + ------                            |\/ 3 *3 /   - + ------             ___       |       /       ____                       
      \/    2     18               1              |        \/    2     18             \/ 3        |     \/    2     18               1              |      \/    2     18             \/ 3        |      /  1   \/ 93              1         
0 + - ----------------- + ------------------- + I*|- ----------------------- - -------------------| + - ----------------- + ------------------- + I*|----------------------- + -------------------| + 3 /   - + ------  - -------------------
              2                  ____________     |             2                     ____________|             2                  ____________     |           2                     ____________|   \/    2     18             ____________
                                /       ____      |                                  /       ____ |                               /       ____      |                                /       ____ |                             /       ____ 
                               /  1   \/ 93       |                                 /  1   \/ 93  |                              /  1   \/ 93       |                               /  1   \/ 93  |                            /  1   \/ 93  
                          6*3 /   - + ------      |                            6*3 /   - + ------ |                         6*3 /   - + ------      |                          6*3 /   - + ------ |                       3*3 /   - + ------ 
                            \/    2     18        \                              \/    2     18   /                           \/    2     18        \                            \/    2     18   /                         \/    2     18   

$$\left(- \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}\right) + \left(\left(0 + \left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)\right)\right)$$

=

  /           ____________                      \     /             ____________                      \
  |          /       ____                       |     |            /       ____                       |
  |  ___    /  1   \/ 93                        |     |    ___    /  1   \/ 93                        |
  |\/ 3 *3 /   - + ------             ___       |     |  \/ 3 *3 /   - + ------             ___       |
  |      \/    2     18             \/ 3        |     |        \/    2     18             \/ 3        |
I*|----------------------- + -------------------| + I*|- ----------------------- - -------------------|
  |           2                     ____________|     |             2                     ____________|
  |                                /       ____ |     |                                  /       ____ |
  |                               /  1   \/ 93  |     |                                 /  1   \/ 93  |
  |                          6*3 /   - + ------ |     |                            6*3 /   - + ------ |
  \                            \/    2     18   /     \                              \/    2     18   /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right) + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)$$

произведение

  /       ____________                           /             ____________                      \\ /       ____________                           /           ____________                      \\                                          
  |      /       ____                            |            /       ____                       || |      /       ____                            |          /       ____                       ||                                          
  |     /  1   \/ 93                             |    ___    /  1   \/ 93                        || |     /  1   \/ 93                             |  ___    /  1   \/ 93                        || /     ____________                      \
  |  3 /   - + ------                            |  \/ 3 *3 /   - + ------             ___       || |  3 /   - + ------                            |\/ 3 *3 /   - + ------             ___       || |    /       ____                       |
  |  \/    2     18               1              |        \/    2     18             \/ 3        || |  \/    2     18               1              |      \/    2     18             \/ 3        || |   /  1   \/ 93              1         |
1*|- ----------------- + ------------------- + I*|- ----------------------- - -------------------||*|- ----------------- + ------------------- + I*|----------------------- + -------------------||*|3 /   - + ------  - -------------------|
  |          2                  ____________     |             2                     ____________|| |          2                  ____________     |           2                     ____________|| |\/    2     18             ____________|
  |                            /       ____      |                                  /       ____ || |                            /       ____      |                                /       ____ || |                          /       ____ |
  |                           /  1   \/ 93       |                                 /  1   \/ 93  || |                           /  1   \/ 93       |                               /  1   \/ 93  || |                         /  1   \/ 93  |
  |                      6*3 /   - + ------      |                            6*3 /   - + ------ || |                      6*3 /   - + ------      |                          6*3 /   - + ------ || |                    3*3 /   - + ------ |
  \                        \/    2     18        \                              \/    2     18   // \                        \/    2     18        \                            \/    2     18   // \                      \/    2     18   /

$$\left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(\frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2}\right)\right) 1 \left(- \frac{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} + \frac{1}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}{2} - \frac{\sqrt{3}}{6 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}}\right)\right) \left(- \frac{1}{3 \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{93}}{18}}\right)$$

=

1

$$1$$

Теорема Виета

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

Численный ответ

x1 = -0.34116390191401 - 1.16154139999725*i

x2 = -0.34116390191401 + 1.16154139999725*i

x3 = 0.682327803828019

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