Решите уравнение x^2+y^2=2019 (х в квадрате плюс у в квадрате равно 2019) - Найдите корень уравнения подробно по-шагам. [Есть ОТВЕТ!]

x^2+y^2=2019 (уравнение)

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    Найду корень уравнения: x^2+y^2=2019

    Решение

    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
    График
    Быстрый ответ [src]
               _____________________________________________                                                            _____________________________________________                                                   
              /                         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 [src]
    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
    $$x_{1} + x_{2} = - p$$
    $$x_{1} x_{2} = q$$
    $$x_{1} + x_{2} = 0$$
    $$x_{1} x_{2} = y^{2} - 2019$$
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