Решение уравнений/ Любые/ (a-1)x^2-2ax+2a-2=0

(a-1)x^2-2ax+2a-2=0 (уравнение)

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    Найду корень уравнения: (a-1)x^2-2ax+2a-2=0

    Решение

    Вы ввели [src]
             2                      
(a - 1)*x  - 2*a*x + 2*a - 2 = 0
    (2a+(−2ax+x2(a−1)))−2=0\left(2 a + \left(- 2 a x + x^{2} \left(a - 1\right)\right)\right) - 2 = 0
    Упростить
    Подробное решение
    Раскроем выражение в уравнении
    (2a+(−2ax+x2(a−1)))−2=0\left(2 a + \left(- 2 a x + x^{2} \left(a - 1\right)\right)\right) - 2 = 0
    Получаем квадратное уравнение
    ax2+2a−2ax−x2−2=0a x^{2} + 2 a - 2 a x - x^{2} - 2 = 0
    Это уравнение вида
    a*x^2 + b*x + c = 0

    Квадратное уравнение можно решить
    с помощью дискриминанта.
    Корни квадратного уравнения:
    x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
    x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
    где D = b^2 - 4*a*c - это дискриминант.
    Т.к.
    False

    b=−2ab = - 2 a
    c=2a−2c = 2 a - 2
    , то
    D = b^2 - 4 * a * c = 

    (-2*a)^2 - 4 * (-1 + a) * (-2 + 2*a) = 4*a^2 - (-4 + 4*a)*(-2 + 2*a)

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

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

    или
    x1=2a+4a2−(2a−2)(4a−4)2a−2x_{1} = \frac{2 a + \sqrt{4 a^{2} - \left(2 a - 2\right) \left(4 a - 4\right)}}{2 a - 2}
    x2=2a−4a2−(2a−2)(4a−4)2a−2x_{2} = \frac{2 a - \sqrt{4 a^{2} - \left(2 a - 2\right) \left(4 a - 4\right)}}{2 a - 2}
    График
    Быстрый ответ [src]
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       |             |  4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |   |  4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |      |                |  4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |   |  4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |      
       |(-1 + re(a))*|- \/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *sin|--------------------------------------------------------------| + im(a)|   |- \/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *cos|--------------------------------------------------------------| + re(a)|*im(a)|   (-1 + re(a))*|- \/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *cos|--------------------------------------------------------------| + re(a)|   |- \/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *sin|--------------------------------------------------------------| + im(a)|*im(a)
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x1 = I*|----------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------| + ----------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------
       |                                                                                  2     2                                                                                                                                                         2     2                                                                     |                                                                                     2     2                                                                                                                                                         2     2                                                                     
       \                                                                      (-1 + re(a))  + im (a)                                                                                                                                          (-1 + re(a))  + im (a)                                                                  /                                                                         (-1 + re(a))  + im (a)                                                                                                                                          (-1 + re(a))  + im (a)
    x1=(−(−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24sin⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+im⁡(a))im⁡(a)(re⁡(a)−1)2+(im⁡(a))2+(−(−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24cos⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+re⁡(a))(re⁡(a)−1)(re⁡(a)−1)2+(im⁡(a))2+i((−(−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24sin⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+im⁡(a))(re⁡(a)−1)(re⁡(a)−1)2+(im⁡(a))2−(−(−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24cos⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+re⁡(a))im⁡(a)(re⁡(a)−1)2+(im⁡(a))2)x_{1} = \frac{\left(- \sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(- \sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(- \sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(- \sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)
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       |             |4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |   |4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |      |                |4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |   |4 /                           2   /       2        2             \      |atan2\4*im(a) - 2*im(a)*re(a), -2 + im (a) - re (a) + 4*re(a)/|        |      
       |(-1 + re(a))*|\/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *sin|--------------------------------------------------------------| + im(a)|   |\/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *cos|--------------------------------------------------------------| + re(a)|*im(a)|   (-1 + re(a))*|\/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *cos|--------------------------------------------------------------| + re(a)|   |\/   (4*im(a) - 2*im(a)*re(a))  + \-2 + im (a) - re (a) + 4*re(a)/  *sin|--------------------------------------------------------------| + im(a)|*im(a)
       |             \                                                                        \                              2                               /        /   \                                                                        \                              2                               /        /      |                \                                                                        \                              2                               /        /   \                                                                        \                              2                               /        /      
x2 = I*|--------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------| + --------------------------------------------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------
       |                                                                                 2     2                                                                                                                                                       2     2                                                                    |                                                                                    2     2                                                                                                                                                       2     2                                                                    
       \                                                                     (-1 + re(a))  + im (a)                                                                                                                                        (-1 + re(a))  + im (a)                                                                 /                                                                        (-1 + re(a))  + im (a)                                                                                                                                        (-1 + re(a))  + im (a)
    x2=((−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24sin⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+im⁡(a))im⁡(a)(re⁡(a)−1)2+(im⁡(a))2+((−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24cos⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+re⁡(a))(re⁡(a)−1)(re⁡(a)−1)2+(im⁡(a))2+i(((−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24sin⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+im⁡(a))(re⁡(a)−1)(re⁡(a)−1)2+(im⁡(a))2−((−2re⁡(a)im⁡(a)+4im⁡(a))2+(−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)24cos⁡(atan2⁡(−2re⁡(a)im⁡(a)+4im⁡(a),−(re⁡(a))2+4re⁡(a)+(im⁡(a))2−2)2)+re⁡(a))im⁡(a)(re⁡(a)−1)2+(im⁡(a))2)x_{2} = \frac{\left(\sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + i \left(\frac{\left(\sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{im}{\left(a\right)}\right) \left(\operatorname{re}{\left(a\right)} - 1\right)}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{\left(\sqrt[4]{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + 4 \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 2 \right)}}{2} \right)} + \operatorname{re}{\left(a\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)} - 1\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right)
    Решение параметрического уравнения
    Дано уравнение с параметром:
    −2ax+2a+x2(a−1)−2=0- 2 a x + 2 a + x^{2} \left(a - 1\right) - 2 = 0
    Коэффициент при x равен
    a−1a - 1
    тогда возможные случаи для a :
    a<1a < 1
    a=1a = 1
    Рассмотри все случаи подробнее:
    При
    a<1a < 1
    уравнение будет
    −x2−2=0- x^{2} - 2 = 0
    его решение
    нет решений
    При
    a=1a = 1
    уравнение будет
    −2x=0- 2 x = 0
    его решение
    x=0x = 0