x(y+4)=3y+15 (уравнение)

Найду корень уравнения: x(y+4)=3y+15

Вы ввели

x*(y + 4) = 3*y + 15

$$x \left(y + 4\right) = 3 y + 15$$

Подробное решение

Дано линейное уравнение:

x*(y+4) = 3*y+15

Раскрываем скобочки в левой части ур-ния

xy+4 = 3*y+15

Разделим обе части ур-ния на 4 + y

x = 15 + 3*y / (4 + y)

Получим ответ: x = 3*(5 + y)/(4 + y)

Быстрый ответ

                                                                      2                                     
       /   3*(5 + re(y))*im(y)     3*(4 + re(y))*im(y) \          3*im (y)         3*(4 + re(y))*(5 + re(y))
x1 = I*|- --------------------- + ---------------------| + --------------------- + -------------------------
       |             2     2                 2     2   |              2     2                   2     2     
       \  (4 + re(y))  + im (y)   (4 + re(y))  + im (y)/   (4 + re(y))  + im (y)     (4 + re(y))  + im (y)  

$$x_{1} = i \left(\frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(y\right)} + 5\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \left(\operatorname{re}{\left(y\right)} + 5\right)}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$

Сумма и произведение корней

сумма

                                                                 2                                     
  /   3*(5 + re(y))*im(y)     3*(4 + re(y))*im(y) \          3*im (y)         3*(4 + re(y))*(5 + re(y))
I*|- --------------------- + ---------------------| + --------------------- + -------------------------
  |             2     2                 2     2   |              2     2                   2     2     
  \  (4 + re(y))  + im (y)   (4 + re(y))  + im (y)/   (4 + re(y))  + im (y)     (4 + re(y))  + im (y)  

$$i \left(\frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(y\right)} + 5\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \left(\operatorname{re}{\left(y\right)} + 5\right)}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$

=

                                                                 2                                     
  /   3*(5 + re(y))*im(y)     3*(4 + re(y))*im(y) \          3*im (y)         3*(4 + re(y))*(5 + re(y))
I*|- --------------------- + ---------------------| + --------------------- + -------------------------
  |             2     2                 2     2   |              2     2                   2     2     
  \  (4 + re(y))  + im (y)   (4 + re(y))  + im (y)/   (4 + re(y))  + im (y)     (4 + re(y))  + im (y)  

$$i \left(\frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(y\right)} + 5\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \left(\operatorname{re}{\left(y\right)} + 5\right)}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$

произведение

                                                                 2                                     
  /   3*(5 + re(y))*im(y)     3*(4 + re(y))*im(y) \          3*im (y)         3*(4 + re(y))*(5 + re(y))
I*|- --------------------- + ---------------------| + --------------------- + -------------------------
  |             2     2                 2     2   |              2     2                   2     2     
  \  (4 + re(y))  + im (y)   (4 + re(y))  + im (y)/   (4 + re(y))  + im (y)     (4 + re(y))  + im (y)  

$$i \left(\frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{3 \left(\operatorname{re}{\left(y\right)} + 5\right) \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{3 \left(\operatorname{re}{\left(y\right)} + 4\right) \left(\operatorname{re}{\left(y\right)} + 5\right)}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{3 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$

=

  /  2                                       \
3*\im (y) + (4 + re(y))*(5 + re(y)) - I*im(y)/
----------------------------------------------
                       2     2                
            (4 + re(y))  + im (y)             

$$\frac{3 \left(\left(\operatorname{re}{\left(y\right)} + 4\right) \left(\operatorname{re}{\left(y\right)} + 5\right) + \left(\operatorname{im}{\left(y\right)}\right)^{2} - i \operatorname{im}{\left(y\right)}\right)}{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$

Решение параметрического уравнения

Дано уравнение с параметром:
$$x \left(y + 4\right) = 3 y + 15$$
Коэффициент при x равен
$$y + 4$$
тогда возможные случаи для y :
$$y < -4$$
$$y = -4$$
Рассмотри все случаи подробнее:
При
$$y < -4$$
уравнение будет
$$- x = 0$$
его решение
$$x = 0$$
При
$$y = -4$$
уравнение будет
$$-3 = 0$$
его решение
нет решений