Подробное решение
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.
$$a = 1$$
$$b = 0$$
$$c = y$$
, то
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (y) = -4*y
Уравнение имеет два корня.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
или
$$x_{1} = \sqrt{- y}$$
Упростить$$x_{2} = - \sqrt{- y}$$
Упростить _________________ _________________
4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\
x1 = - \/ im (y) + re (y) *cos|---------------------| - I*\/ im (y) + re (y) *sin|---------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}$$
_________________ _________________
4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\
x2 = \/ im (y) + re (y) *cos|---------------------| + I*\/ im (y) + re (y) *sin|---------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}$$
Сумма и произведение корней
[src] _________________ _________________ _________________ _________________
4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\
- \/ im (y) + re (y) *cos|---------------------| - I*\/ im (y) + re (y) *sin|---------------------| + \/ im (y) + re (y) *cos|---------------------| + I*\/ im (y) + re (y) *sin|---------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right)$$
$$0$$
/ _________________ _________________ \ / _________________ _________________ \
| 4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\| |4 / 2 2 /atan2(-im(y), -re(y))\ 4 / 2 2 /atan2(-im(y), -re(y))\|
|- \/ im (y) + re (y) *cos|---------------------| - I*\/ im (y) + re (y) *sin|---------------------||*|\/ im (y) + re (y) *cos|---------------------| + I*\/ im (y) + re (y) *sin|---------------------||
\ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}{2} \right)}\right)$$
_________________
/ 2 2 I*atan2(-im(y), -re(y))
-\/ im (y) + re (y) *e
$$- \sqrt{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- \operatorname{im}{\left(y\right)},- \operatorname{re}{\left(y\right)} \right)}}$$