Решите уравнение x^2+t*x+3=0 (х в квадрате плюс t умножить на х плюс 3 равно 0) - Найдите корень уравнения подробно по-шагам. [Есть ОТВЕТ!]

x^2+t*x+3=0 (уравнение)

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    Найду корень уравнения: x^2+t*x+3=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.
    $$a = 1$$
    $$b = t$$
    $$c = 3$$
    , то
    D = b^2 - 4 * a * c = 

    (t)^2 - 4 * (1) * (3) = -12 + t^2

    Уравнение имеет два корня.
    x1 = (-b + sqrt(D)) / (2*a)

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

    или
    $$x_{1} = - \frac{t}{2} + \frac{\sqrt{t^{2} - 12}}{2}$$
    Упростить
    $$x_{2} = - \frac{t}{2} - \frac{\sqrt{t^{2} - 12}}{2}$$
    Упростить
    График
    Быстрый ответ [src]
                     /              ____________________________________________                                                 \       ____________________________________________                                                 
                     |             /                        2                       /     /                       2        2   \\|      /                        2                       /     /                       2        2   \\
                     |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/|
                     |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------|
           re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /
    x1 = - ----- + I*|- ----- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------
             2       \    2                                                     2                                                /                                                   2                                                
    $$x_{1} = i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}$$
                     /              ____________________________________________                                                 \       ____________________________________________                                                 
                     |             /                        2                       /     /                       2        2   \\|      /                        2                       /     /                       2        2   \\
                     |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/|
                     |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------|
           re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /
    x2 = - ----- + I*|- ----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
             2       \    2                                                     2                                                /                                                   2                                                
    $$x_{2} = i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}$$
    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(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/|               |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/|
                |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------|               |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------|
      re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /     re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /
    - ----- + I*|- ----- - -------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------- + - ----- + I*|- ----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
        2       \    2                                                     2                                                /                                                   2                                                       2       \    2                                                     2                                                /                                                   2                                                
    $$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}\right)$$
    =
               /              ____________________________________________                                                 \     /              ____________________________________________                                                 \
               |             /                        2                       /     /                       2        2   \\|     |             /                        2                       /     /                       2        2   \\|
               |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||     |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||
               |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||     |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||
               |  im(t)                                                       \                     2                     /|     |  im(t)                                                       \                     2                     /|
    -re(t) + I*|- ----- + -------------------------------------------------------------------------------------------------| + I*|- ----- - -------------------------------------------------------------------------------------------------|
               \    2                                                     2                                                /     \    2                                                     2                                                /
    $$i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) - \operatorname{re}{\left(t\right)}$$
    произведение
    /            /              ____________________________________________                                                 \       ____________________________________________                                                 \ /            /              ____________________________________________                                                 \       ____________________________________________                                                 \
    |            |             /                        2                       /     /                       2        2   \\|      /                        2                       /     /                       2        2   \\| |            |             /                        2                       /     /                       2        2   \\|      /                        2                       /     /                       2        2   \\|
    |            |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/|| |            |          4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||   4 /  /        2        2   \        2      2        |atan2\2*im(t)*re(t), -12 + re (t) - im (t)/||
    |            |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------|| |            |          \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *sin|-------------------------------------------||   \/   \-12 + re (t) - im (t)/  + 4*im (t)*re (t) *cos|-------------------------------------------||
    |  re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /| |  re(t)     |  im(t)                                                       \                     2                     /|                                                       \                     2                     /|
    |- ----- + I*|- ----- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------|*|- ----- + I*|- ----- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------|
    \    2       \    2                                                     2                                                /                                                   2                                                / \    2       \    2                                                     2                                                /                                                   2                                                /
    $$\left(i \left(- \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) - \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(t\right)}}{2}\right) + \frac{\sqrt[4]{\left(\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12\right)^{2} + 4 \left(\operatorname{re}{\left(t\right)}\right)^{2} \left(\operatorname{im}{\left(t\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(t\right)} \operatorname{im}{\left(t\right)},\left(\operatorname{re}{\left(t\right)}\right)^{2} - \left(\operatorname{im}{\left(t\right)}\right)^{2} - 12 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(t\right)}}{2}\right)$$
    =
    3
    $$3$$
    Теорема Виета
    это приведённое квадратное уравнение
    $$p x + q + x^{2} = 0$$
    где
    $$p = \frac{b}{a}$$
    $$p = t$$
    $$q = \frac{c}{a}$$
    $$q = 3$$
    Формулы Виета
    $$x_{1} + x_{2} = - p$$
    $$x_{1} x_{2} = q$$
    $$x_{1} + x_{2} = - t$$
    $$x_{1} x_{2} = 3$$
    ×

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