x^3-x^2-1=0 (уравнение)

Найду корень уравнения: x^3-x^2-1=0

Вы ввели

 3    2        
x  - x  - 1 = 0

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

Быстрый ответ

              _____________                             /             _____________                        \
             /        ____                              |            /        ____                         |
            /  29   \/ 93                               |    ___    /  29   \/ 93                          |
         3 /   -- + ------                              |  \/ 3 *3 /   -- + ------              ___        |
     1   \/    54     18                1               |        \/    54     18              \/ 3         |
x1 = - - ------------------ - --------------------- + I*|- ------------------------ + ---------------------|
     3           2                    _____________     |             2                       _____________|
                                     /        ____      |                                    /        ____ |
                                    /  29   \/ 93       |                                   /  29   \/ 93  |
                              18*3 /   -- + ------      |                             18*3 /   -- + ------ |
                                 \/    54     18        \                                \/    54     18   /

$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}\right)$$

              _____________                             /           _____________                        \
             /        ____                              |          /        ____                         |
            /  29   \/ 93                               |  ___    /  29   \/ 93                          |
         3 /   -- + ------                              |\/ 3 *3 /   -- + ------              ___        |
     1   \/    54     18                1               |      \/    54     18              \/ 3         |
x2 = - - ------------------ - --------------------- + I*|------------------------ - ---------------------|
     3           2                    _____________     |           2                       _____________|
                                     /        ____      |                                  /        ____ |
                                    /  29   \/ 93       |                                 /  29   \/ 93  |
                              18*3 /   -- + ------      |                           18*3 /   -- + ------ |
                                 \/    54     18        \                              \/    54     18   /

$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2}\right)$$

              _____________                       
             /        ____                        
     1      /  29   \/ 93              1          
x3 = - + 3 /   -- + ------  + --------------------
     3   \/    54     18             _____________
                                    /        ____ 
                                   /  29   \/ 93  
                              9*3 /   -- + ------ 
                                \/    54     18   

$$x_{3} = \frac{1}{9 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}$$

Сумма и произведение корней

сумма

             _____________                             /             _____________                        \            _____________                             /           _____________                        \                                                
            /        ____                              |            /        ____                         |           /        ____                              |          /        ____                         |                                                
           /  29   \/ 93                               |    ___    /  29   \/ 93                          |          /  29   \/ 93                               |  ___    /  29   \/ 93                          |            _____________                       
        3 /   -- + ------                              |  \/ 3 *3 /   -- + ------              ___        |       3 /   -- + ------                              |\/ 3 *3 /   -- + ------              ___        |           /        ____                        
    1   \/    54     18                1               |        \/    54     18              \/ 3         |   1   \/    54     18                1               |      \/    54     18              \/ 3         |   1      /  29   \/ 93              1          
0 + - - ------------------ - --------------------- + I*|- ------------------------ + ---------------------| + - - ------------------ - --------------------- + I*|------------------------ - ---------------------| + - + 3 /   -- + ------  + --------------------
    3           2                    _____________     |             2                       _____________|   3           2                    _____________     |           2                       _____________|   3   \/    54     18             _____________
                                    /        ____      |                                    /        ____ |                                   /        ____      |                                  /        ____ |                                  /        ____ 
                                   /  29   \/ 93       |                                   /  29   \/ 93  |                                  /  29   \/ 93       |                                 /  29   \/ 93  |                                 /  29   \/ 93  
                             18*3 /   -- + ------      |                             18*3 /   -- + ------ |                            18*3 /   -- + ------      |                           18*3 /   -- + ------ |                            9*3 /   -- + ------ 
                                \/    54     18        \                                \/    54     18   /                               \/    54     18        \                              \/    54     18   /                              \/    54     18   

$$\left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}\right) + \left(\left(0 + \left(- \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2}\right)\right)\right)$$

=

      /           _____________                        \     /             _____________                        \
      |          /        ____                         |     |            /        ____                         |
      |  ___    /  29   \/ 93                          |     |    ___    /  29   \/ 93                          |
      |\/ 3 *3 /   -- + ------              ___        |     |  \/ 3 *3 /   -- + ------              ___        |
      |      \/    54     18              \/ 3         |     |        \/    54     18              \/ 3         |
1 + I*|------------------------ - ---------------------| + I*|- ------------------------ + ---------------------|
      |           2                       _____________|     |             2                       _____________|
      |                                  /        ____ |     |                                    /        ____ |
      |                                 /  29   \/ 93  |     |                                   /  29   \/ 93  |
      |                           18*3 /   -- + ------ |     |                             18*3 /   -- + ------ |
      \                              \/    54     18   /     \                                \/    54     18   /

$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}\right) + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2}\right)$$

произведение

  /         _____________                             /             _____________                        \\ /         _____________                             /           _____________                        \\                                                
  |        /        ____                              |            /        ____                         || |        /        ____                              |          /        ____                         ||                                                
  |       /  29   \/ 93                               |    ___    /  29   \/ 93                          || |       /  29   \/ 93                               |  ___    /  29   \/ 93                          || /         _____________                       \
  |    3 /   -- + ------                              |  \/ 3 *3 /   -- + ------              ___        || |    3 /   -- + ------                              |\/ 3 *3 /   -- + ------              ___        || |        /        ____                        |
  |1   \/    54     18                1               |        \/    54     18              \/ 3         || |1   \/    54     18                1               |      \/    54     18              \/ 3         || |1      /  29   \/ 93              1          |
1*|- - ------------------ - --------------------- + I*|- ------------------------ + ---------------------||*|- - ------------------ - --------------------- + I*|------------------------ - ---------------------||*|- + 3 /   -- + ------  + --------------------|
  |3           2                    _____________     |             2                       _____________|| |3           2                    _____________     |           2                       _____________|| |3   \/    54     18             _____________|
  |                                /        ____      |                                    /        ____ || |                                /        ____      |                                  /        ____ || |                               /        ____ |
  |                               /  29   \/ 93       |                                   /  29   \/ 93  || |                               /  29   \/ 93       |                                 /  29   \/ 93  || |                              /  29   \/ 93  |
  |                         18*3 /   -- + ------      |                             18*3 /   -- + ------ || |                         18*3 /   -- + ------      |                           18*3 /   -- + ------ || |                         9*3 /   -- + ------ |
  \                            \/    54     18        \                                \/    54     18   // \                            \/    54     18        \                              \/    54     18   // \                           \/    54     18   /

$$\left(- \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2}\right)\right) 1 \left(- \frac{\sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}}\right)\right) \left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{93}}{18} + \frac{29}{54}}\right)$$

=

1

$$1$$

Теорема Виета

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

Численный ответ

x1 = -0.232785615938384 + 0.792551992515448*i

x2 = -0.232785615938384 - 0.792551992515448*i

x3 = 1.46557123187677

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