ax2+bx+c=0 (уравнение)

Найду корень уравнения: ax2+bx+c=0

Вы ввели

   2              
a*x  + b*x + c = 0

$$c + \left(a x^{2} + b x\right) = 0$$

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.

True

True

True

, то

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

(b)^2 - 4 * (a) * (c) = b^2 - 4*a*c

Уравнение имеет два корня.

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

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

или
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$

Быстрый ответ

       //             ________________________________________________________________                                                                    \         /             ________________________________________________________________                                                                    \      \   /             ________________________________________________________________                                                                    \         /             ________________________________________________________________                                                                    \      
       ||            /                                                              2     /     /                              2        2               \\|         |            /                                                              2     /     /                              2        2               \\|      |   |            /                                                              2     /     /                              2        2               \\|         |            /                                                              2     /     /                              2        2               \\|      
       ||         4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||         |         4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||      |   |         4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||         |         4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||      
       ||-im(b) + \/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *sin|--------------------------------------------------------------||*re(a)   |-re(b) + \/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *cos|--------------------------------------------------------------||*im(a)|   |-im(b) + \/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *sin|--------------------------------------------------------------||*im(a)   |-re(b) + \/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *cos|--------------------------------------------------------------||*re(a)
       |\                                                                                 \                              2                               //         \                                                                                 \                              2                               //      |   \                                                                                 \                              2                               //         \                                                                                 \                              2                               //      
x1 = I*|--------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------| + --------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------
       |                                                                     /  2        2   \                                                                                                                                           /  2        2   \                                                                   |                                                                        /  2        2   \                                                                                                                                           /  2        2   \                                                                   
       \                                                                   2*\im (a) + re (a)/                                                                                                                                         2*\im (a) + re (a)/                                                                   /                                                                      2*\im (a) + re (a)/                                                                                                                                         2*\im (a) + re (a)/                                                                   

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

       //    ________________________________________________________________                                                                            \         /    ________________________________________________________________                                                                            \      \   /    ________________________________________________________________                                                                            \         /    ________________________________________________________________                                                                            \      
       ||   /                                                              2     /     /                              2        2               \\        |         |   /                                                              2     /     /                              2        2               \\        |      |   |   /                                                              2     /     /                              2        2               \\        |         |   /                                                              2     /     /                              2        2               \\        |      
       ||4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|        |         |4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|        |      |   |4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|        |         |4 /                              2   /  2        2               \      |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|        |      
       ||\/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *cos|--------------------------------------------------------------| + re(b)|*im(a)   |\/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *sin|--------------------------------------------------------------| + im(b)|*re(a)|   |\/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *cos|--------------------------------------------------------------| + re(b)|*re(a)   |\/   (-4*im(a*c) + 2*im(b)*re(b))  + \re (b) - im (b) - 4*re(a*c)/  *sin|--------------------------------------------------------------| + im(b)|*im(a)
       |\                                                                        \                              2                               /        /         \                                                                        \                              2                               /        /      |   \                                                                        \                              2                               /        /         \                                                                        \                              2                               /        /      
x2 = I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------
       |                                                                    /  2        2   \                                                                                                                                          /  2        2   \                                                                   |                                                                       /  2        2   \                                                                                                                                          /  2        2   \                                                                   
       \                                                                  2*\im (a) + re (a)/                                                                                                                                        2*\im (a) + re (a)/                                                                   /                                                                     2*\im (a) + re (a)/                                                                                                                                        2*\im (a) + re (a)/                                                                   

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

Решение параметрического уравнения

Дано уравнение с параметром:
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
тогда возможные случаи для a :
$$a < 0$$
$$a = 0$$
Рассмотри все случаи подробнее:
При
$$a < 0$$
уравнение будет
$$b x + c - x^{2} = 0$$
его решение
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
При
$$a = 0$$
уравнение будет
$$b x + c = 0$$
его решение
$$x = - \frac{c}{b}$$