Сумма корней x^2=c

Сумма и произведение корней

сумма

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

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

=

0

$$0$$

произведение

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

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

=

    _________________                       
   /   2        2      I*atan2(im(c), re(c))
-\/  im (c) + re (c) *e                     

$$- \sqrt{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(c\right)},\operatorname{re}{\left(c\right)} \right)}}$$

Теорема Виета

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