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ax^2+bx+c=0 equation

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Numerical solution:

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a*x  + b*x + c = 0
$$c + \left(a x^{2} + b x\right) = 0$$
Detail solution
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
True

True

True

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

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

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$
Vieta's Theorem
rewrite the equation
$$c + \left(a x^{2} + b x\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + b x + c}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{b}{a}$$
$$x_{1} x_{2} = \frac{c}{a}$$
The solution of the parametric equation
Given the equation with a parameter:
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$b x + c - x^{2} = 0$$
its solution
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
With
$$a = 0$$
the equation
$$b x + c = 0$$
its solution
$$x = - \frac{c}{b}$$
The graph
Rapid solution [src]
       //             ________________________________________________________________                                                                    \         /             ________________________________________________________________                                                                    \      \   /             ________________________________________________________________                                                                    \         /             ________________________________________________________________                                                                    \      
       ||            /                                                              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)$$
x2 = -(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c, re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*im(a)/(2*(re(a)^2 + im(a)^2)) - (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + i*(-(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*im(a)/(2*(re(a)^2 + im(a)^2))))