x^3-2*x-5=0 (уравнение)

Найду корень уравнения: x^3-2*x-5=0

Вы ввели

 3              
x  - 2*x - 5 = 0

$$x^{3} - 2 x - 5 = 0$$

Быстрый ответ

            ______________                             /             ______________                        \
           /       ______                              |            /       ______                         |
          /  5   \/ 1929                               |    ___    /  5   \/ 1929                          |
       3 /   - + --------                              |  \/ 3 *3 /   - + --------              ___        |
       \/    2      18                 1               |        \/    2      18               \/ 3         |
x1 = - ------------------- - --------------------- + I*|- ------------------------- + ---------------------|
                2                   ______________     |              2                      ______________|
                                   /       ______      |                                    /       ______ |
                                  /  5   \/ 1929       |                                   /  5   \/ 1929  |
                             3*3 /   - + --------      |                              3*3 /   - + -------- |
                               \/    2      18         \                                \/    2      18    /

$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)$$

            ______________                             /           ______________                        \
           /       ______                              |          /       ______                         |
          /  5   \/ 1929                               |  ___    /  5   \/ 1929                          |
       3 /   - + --------                              |\/ 3 *3 /   - + --------              ___        |
       \/    2      18                 1               |      \/    2      18               \/ 3         |
x2 = - ------------------- - --------------------- + I*|------------------------- - ---------------------|
                2                   ______________     |            2                      ______________|
                                   /       ______      |                                  /       ______ |
                                  /  5   \/ 1929       |                                 /  5   \/ 1929  |
                             3*3 /   - + --------      |                            3*3 /   - + -------- |
                               \/    2      18         \                              \/    2      18    /

$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)$$

          ______________                        
         /       ______                         
        /  5   \/ 1929               2          
x3 = 3 /   - + --------  + ---------------------
     \/    2      18              ______________
                                 /       ______ 
                                /  5   \/ 1929  
                           3*3 /   - + -------- 
                             \/    2      18    

$$x_{3} = \frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}$$

Сумма и произведение корней

сумма

           ______________                             /             ______________                        \          ______________                             /           ______________                        \                                              
          /       ______                              |            /       ______                         |         /       ______                              |          /       ______                         |                                              
         /  5   \/ 1929                               |    ___    /  5   \/ 1929                          |        /  5   \/ 1929                               |  ___    /  5   \/ 1929                          |        ______________                        
      3 /   - + --------                              |  \/ 3 *3 /   - + --------              ___        |     3 /   - + --------                              |\/ 3 *3 /   - + --------              ___        |       /       ______                         
      \/    2      18                 1               |        \/    2      18               \/ 3         |     \/    2      18                 1               |      \/    2      18               \/ 3         |      /  5   \/ 1929               2          
0 + - ------------------- - --------------------- + I*|- ------------------------- + ---------------------| + - ------------------- - --------------------- + I*|------------------------- - ---------------------| + 3 /   - + --------  + ---------------------
               2                   ______________     |              2                      ______________|              2                   ______________     |            2                      ______________|   \/    2      18              ______________
                                  /       ______      |                                    /       ______ |                                 /       ______      |                                  /       ______ |                               /       ______ 
                                 /  5   \/ 1929       |                                   /  5   \/ 1929  |                                /  5   \/ 1929       |                                 /  5   \/ 1929  |                              /  5   \/ 1929  
                            3*3 /   - + --------      |                              3*3 /   - + -------- |                           3*3 /   - + --------      |                            3*3 /   - + -------- |                         3*3 /   - + -------- 
                              \/    2      18         \                                \/    2      18    /                             \/    2      18         \                              \/    2      18    /                           \/    2      18    

$$\left(\frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}\right) - \left(\frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}} - i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right) - i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)\right)$$

=

  /           ______________                        \     /             ______________                        \
  |          /       ______                         |     |            /       ______                         |
  |  ___    /  5   \/ 1929                          |     |    ___    /  5   \/ 1929                          |
  |\/ 3 *3 /   - + --------              ___        |     |  \/ 3 *3 /   - + --------              ___        |
  |      \/    2      18               \/ 3         |     |        \/    2      18               \/ 3         |
I*|------------------------- - ---------------------| + I*|- ------------------------- + ---------------------|
  |            2                      ______________|     |              2                      ______________|
  |                                  /       ______ |     |                                    /       ______ |
  |                                 /  5   \/ 1929  |     |                                   /  5   \/ 1929  |
  |                            3*3 /   - + -------- |     |                              3*3 /   - + -------- |
  \                              \/    2      18    /     \                                \/    2      18    /

$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right) + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)$$

произведение

  /       ______________                             /             ______________                        \\ /       ______________                             /           ______________                        \\                                              
  |      /       ______                              |            /       ______                         || |      /       ______                              |          /       ______                         ||                                              
  |     /  5   \/ 1929                               |    ___    /  5   \/ 1929                          || |     /  5   \/ 1929                               |  ___    /  5   \/ 1929                          || /     ______________                        \
  |  3 /   - + --------                              |  \/ 3 *3 /   - + --------              ___        || |  3 /   - + --------                              |\/ 3 *3 /   - + --------              ___        || |    /       ______                         |
  |  \/    2      18                 1               |        \/    2      18               \/ 3         || |  \/    2      18                 1               |      \/    2      18               \/ 3         || |   /  5   \/ 1929               2          |
1*|- ------------------- - --------------------- + I*|- ------------------------- + ---------------------||*|- ------------------- - --------------------- + I*|------------------------- - ---------------------||*|3 /   - + --------  + ---------------------|
  |           2                   ______________     |              2                      ______________|| |           2                   ______________     |            2                      ______________|| |\/    2      18              ______________|
  |                              /       ______      |                                    /       ______ || |                              /       ______      |                                  /       ______ || |                            /       ______ |
  |                             /  5   \/ 1929       |                                   /  5   \/ 1929  || |                             /  5   \/ 1929       |                                 /  5   \/ 1929  || |                           /  5   \/ 1929  |
  |                        3*3 /   - + --------      |                              3*3 /   - + -------- || |                        3*3 /   - + --------      |                            3*3 /   - + -------- || |                      3*3 /   - + -------- |
  \                          \/    2      18         \                                \/    2      18    // \                          \/    2      18         \                              \/    2      18    // \                        \/    2      18    /

$$\left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)\right) 1 \left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)\right) \left(\frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}\right)$$

=

5

$$5$$

Теорема Виета

это приведённое кубическое уравнение
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2$$
$$v = \frac{d}{a}$$
$$v = -5$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2$$
$$x_{1} x_{2} x_{3} = -5$$

Численный ответ

x1 = 2.09455148154233

x2 = -1.04727574077116 + 1.13593988908893*i

x3 = -1.04727574077116 - 1.13593988908893*i

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