x²+px-16=0 (уравнение)

Найду корень уравнения: x²+px-16=0

Вы ввели

 2               
x  + p*x - 16 = 0

$$\left(p x + x^{2}\right) - 16 = 0$$

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.
$$a = 1$$
$$b = p$$
$$c = -16$$
, то

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

(p)^2 - 4 * (1) * (-16) = 64 + p^2

Уравнение имеет два корня.

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

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

или
$$x_{1} = - \frac{p}{2} + \frac{\sqrt{p^{2} + 64}}{2}$$
Упростить
$$x_{2} = - \frac{p}{2} - \frac{\sqrt{p^{2} + 64}}{2}$$
Упростить

Быстрый ответ

                 /              ___________________________________________                                                \       ___________________________________________                                                
                 |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
                 |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/|
                 |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------|
       re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /
x1 = - ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------
         2       \    2                                                    2                                               /                                                  2                                               

$$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}$$

                 /              ___________________________________________                                                \       ___________________________________________                                                
                 |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
                 |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/|
                 |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------|
       re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /
x2 = - ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
         2       \    2                                                    2                                               /                                                  2                                               

$$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}$$

Сумма и произведение корней

сумма

            /              ___________________________________________                                                \       ___________________________________________                                                               /              ___________________________________________                                                \       ___________________________________________                                                
            |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\               |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\
            |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/|               |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/|
            |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------|               |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------|
  re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /     re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /
- ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - ----------------------------------------------------------------------------------------------- + - ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------
    2       \    2                                                    2                                               /                                                  2                                                      2       \    2                                                    2                                               /                                                  2                                               

$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}\right)$$

=

           /              ___________________________________________                                                \     /              ___________________________________________                                                \
           |             /                       2                       /     /                      2        2   \\|     |             /                       2                       /     /                      2        2   \\|
           |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||     |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||
           |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||     |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||
           |  im(p)                                                      \                    2                     /|     |  im(p)                                                      \                    2                     /|
-re(p) + I*|- ----- + -----------------------------------------------------------------------------------------------| + I*|- ----- - -----------------------------------------------------------------------------------------------|
           \    2                                                    2                                               /     \    2                                                    2                                               /

$$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) - \operatorname{re}{\left(p\right)}$$

произведение

/            /              ___________________________________________                                                \       ___________________________________________                                                \ /            /              ___________________________________________                                                \       ___________________________________________                                                \
|            |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\| |            |             /                       2                       /     /                      2        2   \\|      /                       2                       /     /                      2        2   \\|
|            |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/|| |            |          4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||   4 /  /       2        2   \        2      2        |atan2\2*im(p)*re(p), 64 + re (p) - im (p)/||
|            |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------|| |            |          \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *sin|------------------------------------------||   \/   \64 + re (p) - im (p)/  + 4*im (p)*re (p) *cos|------------------------------------------||
|  re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /| |  re(p)     |  im(p)                                                      \                    2                     /|                                                      \                    2                     /|
|- ----- + I*|- ----- - -----------------------------------------------------------------------------------------------| - -----------------------------------------------------------------------------------------------|*|- ----- + I*|- ----- + -----------------------------------------------------------------------------------------------| + -----------------------------------------------------------------------------------------------|
\    2       \    2                                                    2                                               /                                                  2                                               / \    2       \    2                                                    2                                               /                                                  2                                               /

$$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(p\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64\right)^{2} + 4 \left(\operatorname{re}{\left(p\right)}\right)^{2} \left(\operatorname{im}{\left(p\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(p\right)} \operatorname{im}{\left(p\right)},\left(\operatorname{re}{\left(p\right)}\right)^{2} - \left(\operatorname{im}{\left(p\right)}\right)^{2} + 64 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(p\right)}}{2}\right)$$

=

-16

$$-16$$

Теорема Виета

это приведённое квадратное уравнение
$$p x + q + x^{2} = 0$$
где
$$p = \frac{b}{a}$$

True

$$q = \frac{c}{a}$$
$$q = -16$$
Формулы Виета
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = -16$$