y - y/x = x (уравнение)

Найду корень уравнения: y - y/x = x

Вы ввели

    y    
y - - = x
    x    

$$y - \frac{y}{x} = x$$

Подробное решение

Дано уравнение:
$$y - \frac{y}{x} = x$$
Домножим обе части ур-ния на знаменатели:
и x
получим:
$$x \left(y - \frac{y}{x}\right) = x x$$
$$x y - y = x^{2}$$
Перенесём правую часть уравнения в
левую часть уравнения со знаком минус.

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

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

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

Уравнение имеет два корня.

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

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

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

Быстрый ответ

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

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

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

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