x^3-2*x^2-1=0 (уравнение)

Найду корень уравнения: x^3-2*x^2-1=0

Вы ввели

 3      2        
x  - 2*x  - 1 = 0

$$x^{3} - 2 x^{2} - 1 = 0$$

Быстрый ответ

                                      ______________     /             ______________                        \
                                     /        _____      |            /        _____                         |
                                    /  43   \/ 177       |    ___    /  43   \/ 177                          |
                                 3 /   -- + -------      |  \/ 3 *3 /   -- + -------               ___       |
     2             2             \/    54      18        |        \/    54      18             2*\/ 3        |
x1 = - - --------------------- - ------------------- + I*|- ------------------------- + ---------------------|
     3          ______________            2              |              2                      ______________|
               /        _____                            |                                    /        _____ |
              /  43   \/ 177                             |                                   /  43   \/ 177  |
         9*3 /   -- + -------                            |                              9*3 /   -- + ------- |
           \/    54      18                              \                                \/    54      18   /

$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}\right)$$

                                      ______________     /           ______________                        \
                                     /        _____      |          /        _____                         |
                                    /  43   \/ 177       |  ___    /  43   \/ 177                          |
                                 3 /   -- + -------      |\/ 3 *3 /   -- + -------               ___       |
     2             2             \/    54      18        |      \/    54      18             2*\/ 3        |
x2 = - - --------------------- - ------------------- + I*|------------------------- - ---------------------|
     3          ______________            2              |            2                      ______________|
               /        _____                            |                                  /        _____ |
              /  43   \/ 177                             |                                 /  43   \/ 177  |
         9*3 /   -- + -------                            |                            9*3 /   -- + ------- |
           \/    54      18                              \                              \/    54      18   /

$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2}\right)$$

              ______________                        
             /        _____                         
     2      /  43   \/ 177               4          
x3 = - + 3 /   -- + -------  + ---------------------
     3   \/    54      18             ______________
                                     /        _____ 
                                    /  43   \/ 177  
                               9*3 /   -- + ------- 
                                 \/    54      18   

$$x_{3} = \frac{4}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}$$

Сумма и произведение корней

сумма

                                     ______________     /             ______________                        \                                    ______________     /           ______________                        \                                                  
                                    /        _____      |            /        _____                         |                                   /        _____      |          /        _____                         |                                                  
                                   /  43   \/ 177       |    ___    /  43   \/ 177                          |                                  /  43   \/ 177       |  ___    /  43   \/ 177                          |            ______________                        
                                3 /   -- + -------      |  \/ 3 *3 /   -- + -------               ___       |                               3 /   -- + -------      |\/ 3 *3 /   -- + -------               ___       |           /        _____                         
    2             2             \/    54      18        |        \/    54      18             2*\/ 3        |   2             2             \/    54      18        |      \/    54      18             2*\/ 3        |   2      /  43   \/ 177               4          
0 + - - --------------------- - ------------------- + I*|- ------------------------- + ---------------------| + - - --------------------- - ------------------- + I*|------------------------- - ---------------------| + - + 3 /   -- + -------  + ---------------------
    3          ______________            2              |              2                      ______________|   3          ______________            2              |            2                      ______________|   3   \/    54      18             ______________
              /        _____                            |                                    /        _____ |             /        _____                            |                                  /        _____ |                                   /        _____ 
             /  43   \/ 177                             |                                   /  43   \/ 177  |            /  43   \/ 177                             |                                 /  43   \/ 177  |                                  /  43   \/ 177  
        9*3 /   -- + -------                            |                              9*3 /   -- + ------- |       9*3 /   -- + -------                            |                            9*3 /   -- + ------- |                             9*3 /   -- + ------- 
          \/    54      18                              \                                \/    54      18   /         \/    54      18                              \                              \/    54      18   /                               \/    54      18   

$$\left(\frac{4}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}\right) + \left(\left(0 + \left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2}\right)\right)\right)$$

=

      /           ______________                        \     /             ______________                        \
      |          /        _____                         |     |            /        _____                         |
      |  ___    /  43   \/ 177                          |     |    ___    /  43   \/ 177                          |
      |\/ 3 *3 /   -- + -------               ___       |     |  \/ 3 *3 /   -- + -------               ___       |
      |      \/    54      18             2*\/ 3        |     |        \/    54      18             2*\/ 3        |
2 + I*|------------------------- - ---------------------| + I*|- ------------------------- + ---------------------|
      |            2                      ______________|     |              2                      ______________|
      |                                  /        _____ |     |                                    /        _____ |
      |                                 /  43   \/ 177  |     |                                   /  43   \/ 177  |
      |                            9*3 /   -- + ------- |     |                              9*3 /   -- + ------- |
      \                              \/    54      18   /     \                                \/    54      18   /

$$2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}\right) + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2}\right)$$

произведение

  /                                 ______________     /             ______________                        \\ /                                 ______________     /           ______________                        \\                                                  
  |                                /        _____      |            /        _____                         || |                                /        _____      |          /        _____                         ||                                                  
  |                               /  43   \/ 177       |    ___    /  43   \/ 177                          || |                               /  43   \/ 177       |  ___    /  43   \/ 177                          || /         ______________                        \
  |                            3 /   -- + -------      |  \/ 3 *3 /   -- + -------               ___       || |                            3 /   -- + -------      |\/ 3 *3 /   -- + -------               ___       || |        /        _____                         |
  |2             2             \/    54      18        |        \/    54      18             2*\/ 3        || |2             2             \/    54      18        |      \/    54      18             2*\/ 3        || |2      /  43   \/ 177               4          |
1*|- - --------------------- - ------------------- + I*|- ------------------------- + ---------------------||*|- - --------------------- - ------------------- + I*|------------------------- - ---------------------||*|- + 3 /   -- + -------  + ---------------------|
  |3          ______________            2              |              2                      ______________|| |3          ______________            2              |            2                      ______________|| |3   \/    54      18             ______________|
  |          /        _____                            |                                    /        _____ || |          /        _____                            |                                  /        _____ || |                                /        _____ |
  |         /  43   \/ 177                             |                                   /  43   \/ 177  || |         /  43   \/ 177                             |                                 /  43   \/ 177  || |                               /  43   \/ 177  |
  |    9*3 /   -- + -------                            |                              9*3 /   -- + ------- || |    9*3 /   -- + -------                            |                            9*3 /   -- + ------- || |                          9*3 /   -- + ------- |
  \      \/    54      18                              \                                \/    54      18   // \      \/    54      18                              \                              \/    54      18   // \                            \/    54      18   /

$$\left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2}\right)\right) 1 \left(- \frac{\sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} - \frac{2}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}{2} + \frac{2 \sqrt{3}}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}}\right)\right) \left(\frac{4}{9 \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}} + \frac{2}{3} + \sqrt[3]{\frac{\sqrt{177}}{18} + \frac{43}{54}}\right)$$

=

1

$$1$$

Теорема Виета

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

Численный ответ

x1 = -0.102784715200295 - 0.665456951152813*i

x2 = -0.102784715200295 + 0.665456951152813*i

x3 = 2.20556943040059

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