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x^2-5x+12=0 equation

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Numerical solution:

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The solution

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 2               
x  - 5*x + 12 = 0
$$\left(x^{2} - 5 x\right) + 12 = 0$$
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
$$a = 1$$
$$b = -5$$
$$c = 12$$
, then
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (1) * (12) = -23

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
$$x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 12$$
The graph
Rapid solution [src]
             ____
     5   I*\/ 23 
x1 = - - --------
     2      2    
$$x_{1} = \frac{5}{2} - \frac{\sqrt{23} i}{2}$$
             ____
     5   I*\/ 23 
x2 = - + --------
     2      2    
$$x_{2} = \frac{5}{2} + \frac{\sqrt{23} i}{2}$$
x2 = 5/2 + sqrt(23)*i/2
Sum and product of roots [src]
sum
        ____           ____
5   I*\/ 23    5   I*\/ 23 
- - -------- + - + --------
2      2       2      2    
$$\left(\frac{5}{2} - \frac{\sqrt{23} i}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{23} i}{2}\right)$$
=
5
$$5$$
product
/        ____\ /        ____\
|5   I*\/ 23 | |5   I*\/ 23 |
|- - --------|*|- + --------|
\2      2    / \2      2    /
$$\left(\frac{5}{2} - \frac{\sqrt{23} i}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{23} i}{2}\right)$$
=
12
$$12$$
12
Numerical answer [src]
x1 = 2.5 + 2.39791576165636*i
x2 = 2.5 - 2.39791576165636*i
x2 = 2.5 - 2.39791576165636*i