x^2-y^2=31 (уравнение)

Найду корень уравнения: x^2-y^2=31

Вы ввели

 2    2     
x  - y  = 31

$$x^{2} - y^{2} = 31$$

Подробное решение

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

Уравнение превратится из
$$x^{2} - y^{2} = 31$$
в
$$\left(x^{2} - y^{2}\right) - 31 = 0$$
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.
$$a = 1$$
$$b = 0$$
$$c = - y^{2} - 31$$
, то

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

(0)^2 - 4 * (1) * (-31 - y^2) = 124 + 4*y^2

Уравнение имеет два корня.

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

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

или
$$x_{1} = \frac{\sqrt{4 y^{2} + 124}}{2}$$
Упростить
$$x_{2} = - \frac{\sqrt{4 y^{2} + 124}}{2}$$
Упростить

Быстрый ответ

           ___________________________________________                                                         ___________________________________________                                                
          /                       2                       /     /                      2        2   \\        /                       2                       /     /                      2        2   \\
       4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|
x1 = - \/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------|
                                                          \                    2                     /                                                        \                    2                     /

$$x_{1} = - i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)}$$

         ___________________________________________                                                         ___________________________________________                                                
        /                       2                       /     /                      2        2   \\        /                       2                       /     /                      2        2   \\
     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|
x2 = \/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------|
                                                        \                    2                     /                                                        \                    2                     /

$$x_{2} = i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)}$$

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), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|   4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|
- \/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------| + \/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \31 + 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} + 31\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} + 31 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\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} + 31 \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} + 31\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} + 31 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\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} + 31 \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), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|| |4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/|     4 /  /       2        2   \        2      2        |atan2\2*im(y)*re(y), 31 + re (y) - im (y)/||
|- \/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||*|\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||
\                                                     \                    2                     /                                                        \                    2                     // \                                                   \                    2                     /                                                        \                    2                     //

$$\left(- i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}{2} \right)}\right)$$

=

     ___________________________________________                                              
    /                       2                            /                      2        2   \
   /  /       2        2   \        2      2      I*atan2\2*im(y)*re(y), 31 + re (y) - im (y)/
-\/   \31 + re (y) - im (y)/  + 4*im (y)*re (y) *e                                            

$$- \sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 31 \right)}}$$

Теорема Виета

это приведённое квадратное уравнение
$$p x + q + x^{2} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - y^{2} - 31$$
Формулы Виета
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - y^{2} - 31$$