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x^2-x+5=0

x^2-x+5=0 equation

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Numerical solution:

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The solution

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 2            
x  - x + 5 = 0
$$\left(x^{2} - x\right) + 5 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = 5$$
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1) * (5) = -19

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
$$x_{1} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 5$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 5$$
The graph
Sum and product of roots [src]
sum
        ____           ____
1   I*\/ 19    1   I*\/ 19 
- - -------- + - + --------
2      2       2      2    
$$\left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
1
$$1$$
product
/        ____\ /        ____\
|1   I*\/ 19 | |1   I*\/ 19 |
|- - --------|*|- + --------|
\2      2    / \2      2    /
$$\left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
5
$$5$$
5
Rapid solution [src]
             ____
     1   I*\/ 19 
x1 = - - --------
     2      2    
$$x_{1} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
             ____
     1   I*\/ 19 
x2 = - + --------
     2      2    
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
x2 = 1/2 + sqrt(19)*i/2
Numerical answer [src]
x1 = 0.5 - 2.17944947177034*i
x2 = 0.5 + 2.17944947177034*i
x2 = 0.5 + 2.17944947177034*i
The graph
x^2-x+5=0 equation