x^2+y^2=2019 (уравнение)

Найду корень уравнения: x^2+y^2=2019

Вы ввели

 2    2       
x  + y  = 2019

$$x^{2} + y^{2} = 2019$$

Подробное решение

Перенесём правую часть уравнения в
левую часть уравнения со знаком минус.

Уравнение превратится из
$$x^{2} + y^{2} = 2019$$
в
$$\left(x^{2} + y^{2}\right) - 2019 = 0$$
Это уравнение вида

a*x^2 + b*x + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = 1$$
$$b = 0$$
$$c = y^{2} - 2019$$
, то

D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-2019 + y^2) = 8076 - 4*y^2

Уравнение имеет два корня.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

или
$$x_{1} = \frac{\sqrt{8076 - 4 y^{2}}}{2}$$
Упростить
$$x_{2} = - \frac{\sqrt{8076 - 4 y^{2}}}{2}$$
Упростить

Быстрый ответ

           _____________________________________________                                                            _____________________________________________                                                   
          /                         2                       /     /                         2        2   \\        /                         2                       /     /                         2        2   \\
       4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|
x1 = - \/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| - I*\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|---------------------------------------------|
                                                            \                      2                      /                                                          \                      2                      /

$$x_{1} = - i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019 \right)}}{2} \right)}$$

         _____________________________________________                                                            _____________________________________________                                                   
        /                         2                       /     /                         2        2   \\        /                         2                       /     /                         2        2   \\
     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|
x2 = \/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| + I*\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|---------------------------------------------|
                                                          \                      2                      /                                                          \                      2                      /

$$x_{2} = i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019 \right)}}{2} \right)}$$

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), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|   4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|
- \/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| - I*\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|---------------------------------------------| + \/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| + I*\/   \2019 + im (y) - re (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} + 2019\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} + 2019 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\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} + 2019 \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} + 2019\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} + 2019 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\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} + 2019 \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), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|| |4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/|     4 /  /         2        2   \        2      2        |atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/||
|- \/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| - I*\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|---------------------------------------------||*|\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|---------------------------------------------| + I*\/   \2019 + im (y) - re (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} + 2019\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} + 2019 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\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} + 2019 \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} + 2019\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} + 2019 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\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} + 2019 \right)}}{2} \right)}\right)$$

=

     _____________________________________________                                                 
    /                         2                            /                         2        2   \
   /  /         2        2   \        2      2      I*atan2\-2*im(y)*re(y), 2019 + im (y) - re (y)/
-\/   \2019 + im (y) - re (y)/  + 4*im (y)*re (y) *e                                               

$$- \sqrt{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 2019 \right)}}$$

Теорема Виета

это приведённое квадратное уравнение
$$p x + q + x^{2} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = y^{2} - 2019$$
Формулы Виета
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = y^{2} - 2019$$