x^3=a (уравнение)

Найду корень уравнения: x^3=a

Вы ввели

 3    
x  = a

$$x^{3} = a$$

Быстрый ответ

        _________________                                 _________________                         
     6 /   2        2        /atan2(im(a), re(a))\     6 /   2        2        /atan2(im(a), re(a))\
x1 = \/  im (a) + re (a) *cos|-------------------| + I*\/  im (a) + re (a) *sin|-------------------|
                             \         3         /                             \         3         /

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

       /     _________________                                     _________________                         \      _________________                                     _________________                         
       |  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         /
x2 = I*|- --------------------------------------------- - ---------------------------------------------------| - --------------------------------------------- + ---------------------------------------------------
       \                        2                                                  2                         /                         2                                                  2                         

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

       /     _________________                                     _________________                         \      _________________                                     _________________                         
       |  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         /
x3 = I*|- --------------------------------------------- + ---------------------------------------------------| - --------------------------------------------- - ---------------------------------------------------
       \                        2                                                  2                         /                         2                                                  2                         

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

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))\       \\/  im (a) + re (a)                            /          3*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
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = - a$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = - a$$