x^3+y^2 (уравнение)

Найду корень уравнения: x^3+y^2

Вы ввели

 3    2    
x  + y  = 0

$$x^{3} + y^{2} = 0$$

Подробное решение

Это уравнение вида

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

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = 1$$
$$b = 0$$
$$c = x^{3}$$
, то

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

(0)^2 - 4 * (1) * (x^3) = -4*x^3

Уравнение имеет два корня.

y1 = (-b + sqrt(D)) / (2*a)

y2 = (-b - sqrt(D)) / (2*a)

или
$$y_{1} = \sqrt{- x^{3}}$$
$$y_{2} = - \sqrt{- x^{3}}$$

Быстрый ответ

           ___________________________________________________________                                                                        ___________________________________________________________                                                               
          /                          2                              2     /     /  3          2               3          2         \\        /                          2                              2     /     /  3          2               3          2         \\
       4 /  /  3          2         \    /    3          2         \      |atan2\im (x) - 3*re (x)*im(x), - re (x) + 3*im (x)*re(x)/|     4 /  /  3          2         \    /    3          2         \      |atan2\im (x) - 3*re (x)*im(x), - re (x) + 3*im (x)*re(x)/|
y1 = - \/   \im (x) - 3*re (x)*im(x)/  + \- re (x) + 3*im (x)*re(x)/  *cos|---------------------------------------------------------| - I*\/   \im (x) - 3*re (x)*im(x)/  + \- re (x) + 3*im (x)*re(x)/  *sin|---------------------------------------------------------|
                                                                          \                            2                            /                                                                        \                            2                            /

$$y_{1} = - i \sqrt[4]{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3},- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} - \sqrt[4]{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3},- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)}$$

         ___________________________________________________________                                                                        ___________________________________________________________                                                               
        /                          2                              2     /     /  3          2               3          2         \\        /                          2                              2     /     /  3          2               3          2         \\
     4 /  /  3          2         \    /    3          2         \      |atan2\im (x) - 3*re (x)*im(x), - re (x) + 3*im (x)*re(x)/|     4 /  /  3          2         \    /    3          2         \      |atan2\im (x) - 3*re (x)*im(x), - re (x) + 3*im (x)*re(x)/|
y2 = \/   \im (x) - 3*re (x)*im(x)/  + \- re (x) + 3*im (x)*re(x)/  *cos|---------------------------------------------------------| + I*\/   \im (x) - 3*re (x)*im(x)/  + \- re (x) + 3*im (x)*re(x)/  *sin|---------------------------------------------------------|
                                                                        \                            2                            /                                                                        \                            2                            /

$$y_{2} = i \sqrt[4]{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3},- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)} + \sqrt[4]{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 3 \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{3},- \left(\operatorname{re}{\left(x\right)}\right)^{3} + 3 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} \right)}}{2} \right)}$$