Решите уравнение x^2+p*x-16=0 (х в квадрате плюс p умножить на х минус 16 равно 0) - Найдите корень уравнения подробно по-шагам. [Есть ОТВЕТ!]

x^2+p*x-16=0 (уравнение)

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    Найду корень уравнения: x^2+p*x-16=0

    Решение

    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
    График
    Быстрый ответ [src]
                     /              ___________________________________________                                                \       ___________________________________________                                                
                     |             /                       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 [src]
    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
    $$x_{1} + x_{2} = - p$$
    $$x_{1} x_{2} = q$$
    $$x_{1} + x_{2} = - p$$
    $$x_{1} x_{2} = -16$$
    ×

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