Сумма корней x^2-y^2=8

Сумма и произведение корней

сумма

      __________________________________________                                                        __________________________________________                                                      __________________________________________                                                        __________________________________________                                               
     /                      2                       /     /                     2        2   \\        /                      2                       /     /                     2        2   \\      /                      2                       /     /                     2        2   \\        /                      2                       /     /                     2        2   \\
  4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|     4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|   4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|     4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|
- \/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|-----------------------------------------| - I*\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|-----------------------------------------| + \/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|-----------------------------------------|
                                                    \                    2                    /                                                       \                    2                    /                                                     \                    2                    /                                                       \                    2                    /

$$\left(- i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)}\right)$$

=

0

$$0$$

произведение

/      __________________________________________                                                        __________________________________________                                               \ /    __________________________________________                                                        __________________________________________                                               \
|     /                      2                       /     /                     2        2   \\        /                      2                       /     /                     2        2   \\| |   /                      2                       /     /                     2        2   \\        /                      2                       /     /                     2        2   \\|
|  4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|     4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|| |4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/|     4 /  /      2        2   \        2      2        |atan2\2*im(y)*re(y), 8 + re (y) - im (y)/||
|- \/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|-----------------------------------------| - I*\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|-----------------------------------------||*|\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|-----------------------------------------||
\                                                    \                    2                    /                                                       \                    2                    // \                                                  \                    2                    /                                                       \                    2                    //

$$\left(- i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}{2} \right)}\right)$$

=

     __________________________________________                                             
    /                      2                            /                     2        2   \
   /  /      2        2   \        2      2      I*atan2\2*im(y)*re(y), 8 + re (y) - im (y)/
-\/   \8 + re (y) - im (y)/  + 4*im (y)*re (y) *e                                           

$$- \sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 8 \right)}}$$

Теорема Виета

это приведённое квадратное уравнение
$$p x + q + x^{2} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - y^{2} - 8$$
Формулы Виета
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - y^{2} - 8$$