у=сх^2-х (уравнение)

Найду корень уравнения: у=сх^2-х

Вы ввели

       2    
y = c*x  - x

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

Подробное решение

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

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

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

(1)^2 - 4 * (-c) * (y) = 1 + 4*c*y

Уравнение имеет два корня.

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

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

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

Быстрый ответ

       /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                           
       |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\
       |  |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------|
       |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /
x1 = I*|- ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------
       |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                              
       \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              

$$x_{1} = \frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}$$

       /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                           
       |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\
       |  |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------|
       |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /
x2 = I*|- ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------
       |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                              
       \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              

$$x_{2} = \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}$$

Сумма и произведение корней

сумма

  /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                                /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                           
  |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\     |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\
  |  |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------|     |  |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------|
  |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /     |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /
I*|- ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------ + I*|- ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------
  |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                                   |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                              
  \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                                   \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              

$$\left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) + \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right)$$

=

  /  /       ________________________________                                     \            ________________________________                                           \     /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \         /       ________________________________                                     \      
  |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|     |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|      
  |  |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||     |  |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)
  |  \                                           \               2               //                                                      \               2               /|     |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //         \                                           \               2               //      
I*|- ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------| + I*|- ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------
  |                                    /  2        2   \                                                                   /  2        2   \                              |     |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                      /  2        2   \                                 
  \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /     \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                    2*\im (c) + re (c)/                                 

$$\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) + i \left(- \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right)$$

произведение

/  /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                           \ /  /  /       ________________________________                                     \            ________________________________                                           \   /       ________________________________                                     \            ________________________________                                           \
|  |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\| |  |  |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|   |    4 /                2        2          /atan2(4*im(c*y), 1 + 4*re(c*y))\|         4 /                2        2                /atan2(4*im(c*y), 1 + 4*re(c*y))\|
|  |  |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 - \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------|| |  |  |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*im(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *re(c)*sin|-------------------------------||   |1 + \/  (1 + 4*re(c*y))  + 16*im (c*y) *cos|-------------------------------||*re(c)   \/  (1 + 4*re(c*y))  + 16*im (c*y) *im(c)*sin|-------------------------------||
|  |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /| |  |  \                                           \               2               //                                                      \               2               /|   \                                           \               2               //                                                      \               2               /|
|I*|- ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ - ------------------------------------------------------------------------------|*|I*|- ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------|
|  |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                              | |  |                                    /  2        2   \                                                                   /  2        2   \                              |                                     /  2        2   \                                                                   /  2        2   \                              |
\  \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              / \  \                                  2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /                                   2*\im (c) + re (c)/                                                                 2*\im (c) + re (c)/                              /

$$\left(\frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(- \sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) - \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) \left(\frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} + 1\right) \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)} + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{re}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right) + \frac{\sqrt[4]{\left(4 \operatorname{re}{\left(c y\right)} + 1\right)^{2} + 16 \left(\operatorname{im}{\left(c y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(4 \operatorname{im}{\left(c y\right)},4 \operatorname{re}{\left(c y\right)} + 1 \right)}}{2} \right)} \operatorname{im}{\left(c\right)}}{2 \left(\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}\right)}\right)$$

=

  2                2                  2                  2                                                             
im (c)*re(c*y) - re (c)*re(c*y) + I*im (c)*im(c*y) - I*re (c)*im(c*y) - 2*im(c)*im(c*y)*re(c) + 2*I*im(c)*re(c)*re(c*y)
-----------------------------------------------------------------------------------------------------------------------
                                             4        4          2      2                                              
                                           im (c) + re (c) + 2*im (c)*re (c)                                           

$$\frac{- \left(\operatorname{re}{\left(c\right)}\right)^{2} \operatorname{re}{\left(c y\right)} - i \left(\operatorname{re}{\left(c\right)}\right)^{2} \operatorname{im}{\left(c y\right)} + 2 i \operatorname{re}{\left(c\right)} \operatorname{re}{\left(c y\right)} \operatorname{im}{\left(c\right)} - 2 \operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)} \operatorname{im}{\left(c y\right)} + \operatorname{re}{\left(c y\right)} \left(\operatorname{im}{\left(c\right)}\right)^{2} + i \left(\operatorname{im}{\left(c\right)}\right)^{2} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{4} + 2 \left(\operatorname{re}{\left(c\right)}\right)^{2} \left(\operatorname{im}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{4}}$$

Решение параметрического уравнения

Дано уравнение с параметром:
$$y = c x^{2} - x$$
Коэффициент при x равен
$$- c$$
тогда возможные случаи для c :
$$c < 0$$
$$c = 0$$
Рассмотри все случаи подробнее:
При
$$c < 0$$
уравнение будет
$$x^{2} + x + y = 0$$
его решение
$$x = - \frac{\sqrt{1 - 4 y}}{2} - \frac{1}{2}$$
$$x = \frac{\sqrt{1 - 4 y}}{2} - \frac{1}{2}$$
При
$$c = 0$$
уравнение будет
$$x + y = 0$$
его решение
$$x = - y$$

Теорема Виета

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