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

                                     _______________     /                                    _______________\
                                    /          ____      |                                   /          ____ |
                                   /  27   3*\/ 93       |                           ___    /  27   3*\/ 93  |
                                3 /   -- + --------      |          ___            \/ 3 *3 /   -- + -------- |
                 1              \/    2       2          |        \/ 3                   \/    2       2     |
x1 = - ---------------------- + -------------------- + I*|---------------------- + --------------------------|
              _______________            6               |       _______________               6             |
             /          ____                             |      /          ____                              |
            /  27   3*\/ 93                              |     /  27   3*\/ 93                               |
       2*3 /   -- + --------                             |2*3 /   -- + --------                              |
         \/    2       2                                 \  \/    2       2                                  /

$$x_{1} = - \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)$$

                                     _______________     /                                      _______________\
                                    /          ____      |                                     /          ____ |
                                   /  27   3*\/ 93       |                             ___    /  27   3*\/ 93  |
                                3 /   -- + --------      |            ___            \/ 3 *3 /   -- + -------- |
                 1              \/    2       2          |          \/ 3                   \/    2       2     |
x2 = - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------|
              _______________            6               |         _______________               6             |
             /          ____                             |        /          ____                              |
            /  27   3*\/ 93                              |       /  27   3*\/ 93                               |
       2*3 /   -- + --------                             |  2*3 /   -- + --------                              |
         \/    2       2                                 \    \/    2       2                                  /

$$x_{2} = - \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)$$

                                 _______________
                                /          ____ 
                               /  27   3*\/ 93  
                            3 /   -- + -------- 
              1             \/    2       2     
x3 = -------------------- - --------------------
          _______________            3          
         /          ____                        
        /  27   3*\/ 93                         
     3 /   -- + --------                        
     \/    2       2                            

$$x_{3} = - \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}$$

Сумма и произведение корней

сумма

                                    _______________     /                                    _______________\                                   _______________     /                                      _______________\                               _______________
                                   /          ____      |                                   /          ____ |                                  /          ____      |                                     /          ____ |                              /          ____ 
                                  /  27   3*\/ 93       |                           ___    /  27   3*\/ 93  |                                 /  27   3*\/ 93       |                             ___    /  27   3*\/ 93  |                             /  27   3*\/ 93  
                               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          
            /          ____                             |      /          ____                              |           /          ____                             |        /          ____                              |       /          ____                        
           /  27   3*\/ 93                              |     /  27   3*\/ 93                               |          /  27   3*\/ 93                              |       /  27   3*\/ 93                               |      /  27   3*\/ 93                         
      2*3 /   -- + --------                             |2*3 /   -- + --------                              |     2*3 /   -- + --------                             |  2*3 /   -- + --------                              |   3 /   -- + --------                        
        \/    2       2                                 \  \/    2       2                                  /       \/    2       2                                 \    \/    2       2                                  /   \/    2       2                            

$$\left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right) + \left(\left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)\right) + \left(0 + \left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)\right)\right)\right)$$

=

  /                                    _______________\     /                                      _______________\
  |                                   /          ____ |     |                                     /          ____ |
  |                           ___    /  27   3*\/ 93  |     |                             ___    /  27   3*\/ 93  |
  |          ___            \/ 3 *3 /   -- + -------- |     |            ___            \/ 3 *3 /   -- + -------- |
  |        \/ 3                   \/    2       2     |     |          \/ 3                   \/    2       2     |
I*|---------------------- + --------------------------| + I*|- ---------------------- - --------------------------|
  |       _______________               6             |     |         _______________               6             |
  |      /          ____                              |     |        /          ____                              |
  |     /  27   3*\/ 93                               |     |       /  27   3*\/ 93                               |
  |2*3 /   -- + --------                              |     |  2*3 /   -- + --------                              |
  \  \/    2       2                                  /     \    \/    2       2                                  /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right) + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)$$

произведение

  /                                _______________     /                                    _______________\\ /                                _______________     /                                      _______________\\ /                            _______________\
  |                               /          ____      |                                   /          ____ || |                               /          ____      |                                     /          ____ || |                           /          ____ |
  |                              /  27   3*\/ 93       |                           ___    /  27   3*\/ 93  || |                              /  27   3*\/ 93       |                             ___    /  27   3*\/ 93  || |                          /  27   3*\/ 93  |
  |                           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          |
  |        /          ____                             |      /          ____                              || |        /          ____                             |        /          ____                              || |    /          ____                        |
  |       /  27   3*\/ 93                              |     /  27   3*\/ 93                               || |       /  27   3*\/ 93                              |       /  27   3*\/ 93                               || |   /  27   3*\/ 93                         |
  |  2*3 /   -- + --------                             |2*3 /   -- + --------                              || |  2*3 /   -- + --------                             |  2*3 /   -- + --------                              || |3 /   -- + --------                        |
  \    \/    2       2                                 \  \/    2       2                                  // \    \/    2       2                                 \    \/    2       2                                  // \\/    2       2                            /

$$1 \left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(\frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6}\right)\right) \left(- \frac{1}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{6} - \frac{\sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}{3} + \frac{1}{\sqrt[3]{\frac{27}{2} + \frac{3 \sqrt{93}}{2}}}\right)$$

=

          _________________             _______________                        2/3             _________________                      2/3                  _______________                               2/3                                  2/3
       3 /            ____       2/3 3 /          ____    3 ___ /         ____\        ____ 3 /            ____    3 ____ /      ____\       2/3   ____ 3 /          ____    3 ___   ____ /         ____\      3 ___ 6 ___   ____ /      ____\   
     9*\/  108 + 12*\/ 93     9*2   *\/  27 + 3*\/ 93     \/ 2 *\27 + 3*\/ 93 /      \/ 93 *\/  108 + 12*\/ 93     \/ 18 *\9 + \/ 93 /      2   *\/ 93 *\/  27 + 3*\/ 93     \/ 2 *\/ 93 *\27 + 3*\/ 93 /      \/ 2 *\/ 3 *\/ 31 *\9 + \/ 93 /   
-1 + ---------------------- - ------------------------- - ------------------------ - --------------------------- + ---------------------- + ------------------------------ + ------------------------------- - ----------------------------------
               32                         32                         8                            32                         8                            32                                72                                 24                

$$- \frac{\sqrt[3]{2} \cdot \sqrt[6]{3} \sqrt{31} \left(9 + \sqrt{93}\right)^{\frac{2}{3}}}{24} - \frac{\sqrt[3]{2} \left(27 + 3 \sqrt{93}\right)^{\frac{2}{3}}}{8} - \frac{\sqrt{93} \sqrt[3]{108 + 12 \sqrt{93}}}{32} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt[3]{27 + 3 \sqrt{93}}}{32} - 1 + \frac{9 \sqrt[3]{108 + 12 \sqrt{93}}}{32} + \frac{2^{\frac{2}{3}} \sqrt{93} \sqrt[3]{27 + 3 \sqrt{93}}}{32} + \frac{\sqrt[3]{18} \left(9 + \sqrt{93}\right)^{\frac{2}{3}}}{8} + \frac{\sqrt[3]{2} \sqrt{93} \left(27 + 3 \sqrt{93}\right)^{\frac{2}{3}}}{72}$$

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 = -0.682327803828019

x2 = 0.34116390191401 + 1.16154139999725*i

x3 = 0.34116390191401 - 1.16154139999725*i

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