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x^2-5*x+12=0

x^2-5*x+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
(x2−5x)+12=0\left(x^{2} - 5 x\right) + 12 = 0
Simplify
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=−5b = -5
c=12c = 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
x1=52+23i2x_{1} = \frac{5}{2} + \frac{\sqrt{23} i}{2}
x2=52−23i2x_{2} = \frac{5}{2} - \frac{\sqrt{23} i}{2}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−5p = -5
q=caq = \frac{c}{a}
q=12q = 12
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=5x_{1} + x_{2} = 5
x1x2=12x_{1} x_{2} = 12
The graph
-1.00.01.02.03.04.05.06.07.08.09.0020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
        ____           ____
5   I*\/ 23    5   I*\/ 23 
- - -------- + - + --------
2      2       2      2
(52−23i2)+(52+23i2)\left(\frac{5}{2} - \frac{\sqrt{23} i}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{23} i}{2}\right) 
=
5
55 
product
/        ____\ /        ____\
|5   I*\/ 23 | |5   I*\/ 23 |
|- - --------|*|- + --------|
\2      2    / \2      2    /
(52−23i2)(52+23i2)\left(\frac{5}{2} - \frac{\sqrt{23} i}{2}\right) \left(\frac{5}{2} + \frac{\sqrt{23} i}{2}\right) 
=
12
1212 
12
Rapid solution [src] 
             ____
     5   I*\/ 23 
x1 = - - --------
     2      2
x1=52−23i2x_{1} = \frac{5}{2} - \frac{\sqrt{23} i}{2} 
             ____
     5   I*\/ 23 
x2 = - + --------
     2      2
x2=52+23i2x_{2} = \frac{5}{2} + \frac{\sqrt{23} i}{2} 
x2 = 5/2 + sqrt(23)*i/2
Numerical answer [src] 
x1 = 2.5 + 2.39791576165636*i
x2 = 2.5 - 2.39791576165636*i
x2 = 2.5 - 2.39791576165636*i
The graph
x^2-5*x+12=0 equation
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