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*\/ 69       |                             ___    /  27   3*\/ 69  |
                              3 /   -- + --------      |            ___            \/ 3 *3 /   -- + -------- |
               1              \/    2       2          |          \/ 3                   \/    2       2     |
x1 = ---------------------- + -------------------- + I*|- ---------------------- + --------------------------|
            _______________            6               |         _______________               6             |
           /          ____                             |        /          ____                              |
          /  27   3*\/ 69                              |       /  27   3*\/ 69                               |
     2*3 /   -- + --------                             |  2*3 /   -- + --------                              |
       \/    2       2                                 \    \/    2       2                                  /

$$x_{1} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)$$

                                   _______________     /                                    _______________\
                                  /          ____      |                                   /          ____ |
                                 /  27   3*\/ 69       |                           ___    /  27   3*\/ 69  |
                              3 /   -- + --------      |          ___            \/ 3 *3 /   -- + -------- |
               1              \/    2       2          |        \/ 3                   \/    2       2     |
x2 = ---------------------- + -------------------- + I*|---------------------- - --------------------------|
            _______________            6               |       _______________               6             |
           /          ____                             |      /          ____                              |
          /  27   3*\/ 69                              |     /  27   3*\/ 69                               |
     2*3 /   -- + --------                             |2*3 /   -- + --------                              |
       \/    2       2                                 \  \/    2       2                                  /

$$x_{2} = \frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)$$

                                   _______________
                                  /          ____ 
                                 /  27   3*\/ 69  
                              3 /   -- + -------- 
                1             \/    2       2     
x3 = - -------------------- - --------------------
            _______________            3          
           /          ____                        
          /  27   3*\/ 69                         
       3 /   -- + --------                        
       \/    2       2                            

$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}$$

Сумма и произведение корней

сумма

                                  _______________     /                                      _______________\                                 _______________     /                                    _______________\                                 _______________
                                 /          ____      |                                     /          ____ |                                /          ____      |                                   /          ____ |                                /          ____ 
                                /  27   3*\/ 69       |                             ___    /  27   3*\/ 69  |                               /  27   3*\/ 69       |                           ___    /  27   3*\/ 69  |                               /  27   3*\/ 69  
                             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*\/ 69                              |       /  27   3*\/ 69                               |        /  27   3*\/ 69                              |     /  27   3*\/ 69                               |        /  27   3*\/ 69                         
    2*3 /   -- + --------                             |  2*3 /   -- + --------                              |   2*3 /   -- + --------                             |2*3 /   -- + --------                              |     3 /   -- + --------                        
      \/    2       2                                 \    \/    2       2                                  /     \/    2       2                                 \  \/    2       2                                  /     \/    2       2                            

$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right) + \left(\left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)\right) + \left(0 + \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)\right)\right)\right)$$

=

  /                                    _______________\     /                                      _______________\
  |                                   /          ____ |     |                                     /          ____ |
  |                           ___    /  27   3*\/ 69  |     |                             ___    /  27   3*\/ 69  |
  |          ___            \/ 3 *3 /   -- + -------- |     |            ___            \/ 3 *3 /   -- + -------- |
  |        \/ 3                   \/    2       2     |     |          \/ 3                   \/    2       2     |
I*|---------------------- - --------------------------| + I*|- ---------------------- + --------------------------|
  |       _______________               6             |     |         _______________               6             |
  |      /          ____                              |     |        /          ____                              |
  |     /  27   3*\/ 69                               |     |       /  27   3*\/ 69                               |
  |2*3 /   -- + --------                              |     |  2*3 /   -- + --------                              |
  \  \/    2       2                                  /     \    \/    2       2                                  /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)$$

произведение

  /                              _______________     /                                      _______________\\ /                              _______________     /                                    _______________\\ /                              _______________\
  |                             /          ____      |                                     /          ____ || |                             /          ____      |                                   /          ____ || |                             /          ____ |
  |                            /  27   3*\/ 69       |                             ___    /  27   3*\/ 69  || |                            /  27   3*\/ 69       |                           ___    /  27   3*\/ 69  || |                            /  27   3*\/ 69  |
  |                         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*\/ 69                              |       /  27   3*\/ 69                               || |     /  27   3*\/ 69                              |     /  27   3*\/ 69                               || |     /  27   3*\/ 69                         |
  |2*3 /   -- + --------                             |  2*3 /   -- + --------                              || |2*3 /   -- + --------                             |2*3 /   -- + --------                              || |  3 /   -- + --------                        |
  \  \/    2       2                                 \    \/    2       2                                  // \  \/    2       2                                 \  \/    2       2                                  // \  \/    2       2                            /

$$1 \cdot \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6}\right)\right) \left(\frac{1}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{6} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}{3} - \frac{1}{\sqrt[3]{\frac{3 \sqrt{69}}{2} + \frac{27}{2}}}\right)$$

=

          _________________             _______________                      2/3             _________________                        2/3                               2/3                  _______________                                  2/3
       3 /            ____       2/3 3 /          ____    3 ____ /      ____\        ____ 3 /            ____    3 ___ /         ____\      3 ___   ____ /         ____\       2/3   ____ 3 /          ____    3 ___ 6 ___   ____ /      ____\   
     9*\/  108 + 12*\/ 69     9*2   *\/  27 + 3*\/ 69     \/ 18 *\9 + \/ 69 /      \/ 69 *\/  108 + 12*\/ 69     \/ 2 *\27 + 3*\/ 69 /      \/ 2 *\/ 69 *\27 + 3*\/ 69 /      2   *\/ 69 *\/  27 + 3*\/ 69     \/ 2 *\/ 3 *\/ 23 *\9 + \/ 69 /   
-1 + ---------------------- - ------------------------- - ---------------------- - --------------------------- + ------------------------ - ------------------------------- + ------------------------------ + ----------------------------------
               32                         32                        8                           32                          8                              72                               32                                 24                

$$- \frac{\sqrt[3]{18} \left(\sqrt{69} + 9\right)^{\frac{2}{3}}}{8} - \frac{\sqrt[3]{2} \sqrt{69} \left(3 \sqrt{69} + 27\right)^{\frac{2}{3}}}{72} - \frac{9 \cdot 2^{\frac{2}{3}} \sqrt[3]{3 \sqrt{69} + 27}}{32} - \frac{\sqrt{69} \sqrt[3]{12 \sqrt{69} + 108}}{32} - 1 + \frac{2^{\frac{2}{3}} \sqrt{69} \sqrt[3]{3 \sqrt{69} + 27}}{32} + \frac{9 \sqrt[3]{12 \sqrt{69} + 108}}{32} + \frac{\sqrt[3]{2} \sqrt{23} \cdot \sqrt[6]{3} \left(\sqrt{69} + 9\right)^{\frac{2}{3}}}{24} + \frac{\sqrt[3]{2} \left(3 \sqrt{69} + 27\right)^{\frac{2}{3}}}{8}$$

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 = 0.662358978622373 + 0.562279512062301*i

x2 = -1.32471795724475

x3 = 0.662358978622373 - 0.562279512062301*i

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