z^3+a=0 (уравнение)

Найду корень уравнения: z^3+a=0

Вы ввели

 3        
z  + a = 0

$$a + z^{3} = 0$$

Быстрый ответ

        _________________                                   _________________                           
     6 /   2        2        /atan2(-im(a), -re(a))\     6 /   2        2        /atan2(-im(a), -re(a))\
z1 = \/  im (a) + re (a) *cos|---------------------| + I*\/  im (a) + re (a) *sin|---------------------|
                             \          3          /                             \          3          /

$$z_{1} = i \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}$$

       /     _________________                                       _________________                           \      _________________                                       _________________                           
       |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\
       |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------|
       |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /
z2 = I*|- ----------------------------------------------- - -----------------------------------------------------| - ----------------------------------------------- + -----------------------------------------------------
       \                         2                                                    2                          /                          2                                                    2                          

$$z_{2} = i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}$$

       /     _________________                                       _________________                           \      _________________                                       _________________                           
       |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\
       |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------|
       |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /
z3 = I*|- ----------------------------------------------- + -----------------------------------------------------| - ----------------------------------------------- - -----------------------------------------------------
       \                         2                                                    2                          /                          2                                                    2                          

$$z_{3} = i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}$$

Сумма и произведение корней

сумма

                                                                                                        /     _________________                                       _________________                           \      _________________                                       _________________                                /     _________________                                       _________________                           \      _________________                                       _________________                           
                                                                                                        |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\     |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\
   _________________                                   _________________                                |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------|     |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------|
6 /   2        2        /atan2(-im(a), -re(a))\     6 /   2        2        /atan2(-im(a), -re(a))\     |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /     |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /
\/  im (a) + re (a) *cos|---------------------| + I*\/  im (a) + re (a) *sin|---------------------| + I*|- ----------------------------------------------- - -----------------------------------------------------| - ----------------------------------------------- + ----------------------------------------------------- + I*|- ----------------------------------------------- + -----------------------------------------------------| - ----------------------------------------------- - -----------------------------------------------------
                        \          3          /                             \          3          /     \                         2                                                    2                          /                          2                                                    2                               \                         2                                                    2                          /                          2                                                    2                          

$$\left(\left(i \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}\right) + \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right)\right) + \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right)$$

=

  /     _________________                                       _________________                           \     /     _________________                                       _________________                           \                                                    
  |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|     |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|                                                    
  |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||     |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||        _________________                           
  |                          \          3          /                                 \          3          /|     |                          \          3          /                                 \          3          /|     6 /   2        2        /atan2(-im(a), -re(a))\
I*|- ----------------------------------------------- + -----------------------------------------------------| + I*|- ----------------------------------------------- - -----------------------------------------------------| + I*\/  im (a) + re (a) *sin|---------------------|
  \                         2                                                    2                          /     \                         2                                                    2                          /                             \          3          /

$$i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) + i \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}$$

произведение

                                                                                                      /  /     _________________                                       _________________                           \      _________________                                       _________________                           \ /  /     _________________                                       _________________                           \      _________________                                       _________________                           \
                                                                                                      |  |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\| |  |  6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|   6 /   2        2        /atan2(-im(a), -re(a))\     ___ 6 /   2        2        /atan2(-im(a), -re(a))\|
/   _________________                                   _________________                           \ |  |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------|| |  |  \/  im (a) + re (a) *sin|---------------------|   \/ 3 *\/  im (a) + re (a) *cos|---------------------||   \/  im (a) + re (a) *cos|---------------------|   \/ 3 *\/  im (a) + re (a) *sin|---------------------||
|6 /   2        2        /atan2(-im(a), -re(a))\     6 /   2        2        /atan2(-im(a), -re(a))\| |  |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /| |  |                          \          3          /                                 \          3          /|                           \          3          /                                 \          3          /|
|\/  im (a) + re (a) *cos|---------------------| + I*\/  im (a) + re (a) *sin|---------------------||*|I*|- ----------------------------------------------- - -----------------------------------------------------| - ----------------------------------------------- + -----------------------------------------------------|*|I*|- ----------------------------------------------- + -----------------------------------------------------| - ----------------------------------------------- - -----------------------------------------------------|
\                        \          3          /                             \          3          // \  \                         2                                                    2                          /                          2                                                    2                          / \  \                         2                                                    2                          /                          2                                                    2                          /

$$\left(i \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} + \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}\right) \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) \left(i \left(- \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right) - \frac{\sqrt{3} \sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2} - \frac{\sqrt[6]{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{2}\right)$$

=

                     /                                                                                                                          /         im(a)              /atan2(-im(a), -re(a))\\\
                     |                                                                                                                      3*I*|- -------------------- + sin|---------------------|||
                     |                                   /atan2(-im(a), -re(a))\                                                                |     _________________      \          3          /||
   _________________ |                              3*cos|---------------------|                                                                |    /   2        2                                 ||
  /   2        2     |   3/atan2(-im(a), -re(a))\        \          3          /        3/atan2(-im(a), -re(a))\          3*re(a)               \  \/  im (a) + re (a)                              /|
\/  im (a) + re (a) *|cos |---------------------| - ---------------------------- - I*sin |---------------------| - ---------------------- + ---------------------------------------------------------|
                     |    \          3          /                4                       \          3          /        _________________                               4                            |
                     |                                                                                                 /   2        2                                                                |
                     \                                                                                             4*\/  im (a) + re (a)                                                             /

$$\sqrt{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \left(\frac{3 i \left(\sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} - \frac{\operatorname{im}{\left(a\right)}}{\sqrt{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}}\right)}{4} - i \sin^{3}{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} + \cos^{3}{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)} - \frac{3 \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},- \operatorname{re}{\left(a\right)} \right)}}{3} \right)}}{4} - \frac{3 \operatorname{re}{\left(a\right)}}{4 \sqrt{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}}\right)$$

Теорема Виета

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