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(6237x^2+411x+250)/6237=0 equation

A equation with variable:

Numerical solution:

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The solution

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      2                  
6237*x  + 411*x + 250    
--------------------- = 0
         6237            
$$\frac{\left(6237 x^{2} + 411 x\right) + 250}{6237} = 0$$
Detail solution
Expand the expression in the equation
$$\frac{\left(6237 x^{2} + 411 x\right) + 250}{6237} = 0$$
We get the quadratic equation
$$x^{2} + \frac{137 x}{2079} + \frac{250}{6237} = 0$$
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
$$a = 1$$
$$b = \frac{137}{2079}$$
$$c = \frac{250}{6237}$$
, then
D = b^2 - 4 * a * c = 

(137/2079)^2 - 4 * (1) * (250/6237) = -674231/4322241

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)

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

or
$$x_{1} = - \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}$$
$$x_{2} = - \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{137}{2079}$$
$$q = \frac{c}{a}$$
$$q = \frac{250}{6237}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{137}{2079}$$
$$x_{1} x_{2} = \frac{250}{6237}$$
The graph
Sum and product of roots [src]
sum
             ________                ________
  137    I*\/ 674231      137    I*\/ 674231 
- ---- - ------------ + - ---- + ------------
  4158       4158         4158       4158    
$$\left(- \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}\right) + \left(- \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}\right)$$
=
-137 
-----
 2079
$$- \frac{137}{2079}$$
product
/             ________\ /             ________\
|  137    I*\/ 674231 | |  137    I*\/ 674231 |
|- ---- - ------------|*|- ---- + ------------|
\  4158       4158    / \  4158       4158    /
$$\left(- \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}\right) \left(- \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}\right)$$
=
250 
----
6237
$$\frac{250}{6237}$$
250/6237
Rapid solution [src]
                  ________
       137    I*\/ 674231 
x1 = - ---- - ------------
       4158       4158    
$$x_{1} = - \frac{137}{4158} - \frac{\sqrt{674231} i}{4158}$$
                  ________
       137    I*\/ 674231 
x2 = - ---- + ------------
       4158       4158    
$$x_{2} = - \frac{137}{4158} + \frac{\sqrt{674231} i}{4158}$$
x2 = -137/4158 + sqrt(674231)*i/4158
Numerical answer [src]
x1 = -0.0329485329485329 - 0.197478524384922*i
x2 = -0.0329485329485329 + 0.197478524384922*i
x2 = -0.0329485329485329 + 0.197478524384922*i
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