(2ax+b)^2 =D (уравнение)

Найду корень уравнения: (2ax+b)^2 =D

Вы ввели

           2    
(2*a*x + b)  = d

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

Подробное решение

Перенесём правую часть уравнения в
левую часть уравнения со знаком минус.

Уравнение превратится из
$$\left(2 a x + b\right)^{2} = d$$
в
$$- d + \left(2 a x + b\right)^{2} = 0$$
Раскроем выражение в уравнении
$$- d + \left(2 a x + b\right)^{2} = 0$$
Получаем квадратное уравнение
$$4 a^{2} x^{2} + 4 a b x + b^{2} - d = 0$$
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.
$$a = 4 a^{2}$$
$$b = 4 a b$$
$$c = b^{2} - d$$
, то

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

(4*a*b)^2 - 4 * (4*a^2) * (b^2 - d) = -16*a^2*(b^2 - d) + 16*a^2*b^2

Уравнение имеет два корня.

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

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

или
$$x_{1} = \frac{- 4 a b + \sqrt{16 a^{2} b^{2} - 16 a^{2} \left(b^{2} - d\right)}}{8 a^{2}}$$
$$x_{2} = \frac{- 4 a b - \sqrt{16 a^{2} b^{2} - 16 a^{2} \left(b^{2} - d\right)}}{8 a^{2}}$$

Быстрый ответ

       //            _________________                         \         /            _________________                         \      \   /            _________________                         \         /            _________________                         \      
       ||         4 /   2        2        /atan2(im(d), re(d))\|         |         4 /   2        2        /atan2(im(d), re(d))\|      |   |         4 /   2        2        /atan2(im(d), re(d))\|         |         4 /   2        2        /atan2(im(d), re(d))\|      
       ||-im(b) + \/  im (d) + re (d) *sin|-------------------||*re(a)   |-re(b) + \/  im (d) + re (d) *cos|-------------------||*im(a)|   |-im(b) + \/  im (d) + re (d) *sin|-------------------||*im(a)   |-re(b) + \/  im (d) + re (d) *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)/                      

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

       //   _________________                                 \         /   _________________                                 \      \   /   _________________                                 \         /   _________________                                 \      
       ||4 /   2        2        /atan2(im(d), re(d))\        |         |4 /   2        2        /atan2(im(d), re(d))\        |      |   |4 /   2        2        /atan2(im(d), re(d))\        |         |4 /   2        2        /atan2(im(d), re(d))\        |      
       ||\/  im (d) + re (d) *cos|-------------------| + re(b)|*im(a)   |\/  im (d) + re (d) *sin|-------------------| + im(b)|*re(a)|   |\/  im (d) + re (d) *cos|-------------------| + re(b)|*re(a)   |\/  im (d) + re (d) *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)/                     

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