8х²+5х_4=0 (уравнение)

Найду корень уравнения: 8х²+5х_4=0

Вы ввели

   2            
8*x  + 5*x_4 = 0

$$8 x^{2} + 5 x_{4} = 0$$

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.
$$a = 8$$
$$b = 0$$
$$c = 5 x_{4}$$
, то

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

(0)^2 - 4 * (8) * (5*x_4) = -160*x_4

Уравнение имеет два корня.

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

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

или
$$x_{1} = \frac{\sqrt{10} \sqrt{- x_{4}}}{4}$$
Упростить
$$x_{2} = - \frac{\sqrt{10} \sqrt{- x_{4}}}{4}$$
Упростить

Быстрый ответ

                 _____________________                                              _____________________                               
         ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\
       \/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------|
                                          \            2            /                                        \            2            /
x1 = - -------------------------------------------------------------- - ----------------------------------------------------------------
                                     4                                                                 4                                

$$x_{1} = - \frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} - \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}$$

               _____________________                                              _____________________                               
       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\
     \/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------|
                                        \            2            /                                        \            2            /
x2 = -------------------------------------------------------------- + ----------------------------------------------------------------
                                   4                                                                 4                                

$$x_{2} = \frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} + \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}$$

Сумма и произведение корней

сумма

            _____________________                                              _____________________                                            _____________________                                              _____________________                               
    ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\     ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\
  \/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------|   \/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------|
                                     \            2            /                                        \            2            /                                      \            2            /                                        \            2            /
- -------------------------------------------------------------- - ---------------------------------------------------------------- + -------------------------------------------------------------- + ----------------------------------------------------------------
                                4                                                                 4                                                                 4                                                                 4                                

$$\left(- \frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} - \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}\right) + \left(\frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} + \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}\right)$$

=

0

$$0$$

произведение

/            _____________________                                              _____________________                               \ /          _____________________                                              _____________________                               \
|    ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\| |  ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\       ____ 4 /   2          2          /atan2(-im(x_4), -re(x_4))\|
|  \/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------|| |\/ 10 *\/  im (x_4) + re (x_4) *cos|-------------------------|   I*\/ 10 *\/  im (x_4) + re (x_4) *sin|-------------------------||
|                                     \            2            /                                        \            2            /| |                                   \            2            /                                        \            2            /|
|- -------------------------------------------------------------- - ----------------------------------------------------------------|*|-------------------------------------------------------------- + ----------------------------------------------------------------|
\                                4                                                                 4                                / \                              4                                                                 4                                /

$$\left(- \frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} - \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}\right) \left(\frac{\sqrt{10} i \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4} + \frac{\sqrt{10} \sqrt[4]{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}{2} \right)}}{4}\right)$$

=

      _____________________                             
     /   2          2        I*atan2(-im(x_4), -re(x_4))
-5*\/  im (x_4) + re (x_4) *e                           
--------------------------------------------------------
                           8                            

$$- \frac{5 \sqrt{\left(\operatorname{re}{\left(x_{4}\right)}\right)^{2} + \left(\operatorname{im}{\left(x_{4}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- \operatorname{im}{\left(x_{4}\right)},- \operatorname{re}{\left(x_{4}\right)} \right)}}}{8}$$

Теорема Виета

перепишем уравнение
$$8 x^{2} + 5 x_{4} = 0$$
из
$$a x^{2} + b x + c = 0$$
как приведённое квадратное уравнение
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{5 x_{4}}{8} = 0$$
$$p x + q + x^{2} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{5 x_{4}}{8}$$
Формулы Виета
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{5 x_{4}}{8}$$