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x^2-3*x+3=0

x^2-3*x+3=0 equation

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Numerical solution:

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The solution

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 2              
x  - 3*x + 3 = 0
(x2−3x)+3=0\left(x^{2} - 3 x\right) + 3 = 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=−3b = -3
c=3c = 3
, then
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (1) * (3) = -3

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=32+3i2x_{1} = \frac{3}{2} + \frac{\sqrt{3} i}{2}
x2=32−3i2x_{2} = \frac{3}{2} - \frac{\sqrt{3} 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=−3p = -3
q=caq = \frac{c}{a}
q=3q = 3
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=3x_{1} + x_{2} = 3
x1x2=3x_{1} x_{2} = 3
The graph
012345678-4-3-2-1020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
             ___
     3   I*\/ 3 
x1 = - - -------
     2      2
x1=32−3i2x_{1} = \frac{3}{2} - \frac{\sqrt{3} i}{2} 
             ___
     3   I*\/ 3 
x2 = - + -------
     2      2
x2=32+3i2x_{2} = \frac{3}{2} + \frac{\sqrt{3} i}{2} 
x2 = 3/2 + sqrt(3)*i/2
Sum and product of roots [src] 
sum
        ___           ___
3   I*\/ 3    3   I*\/ 3 
- - ------- + - + -------
2      2      2      2
(32−3i2)+(32+3i2)\left(\frac{3}{2} - \frac{\sqrt{3} i}{2}\right) + \left(\frac{3}{2} + \frac{\sqrt{3} i}{2}\right) 
=
3
33 
product
/        ___\ /        ___\
|3   I*\/ 3 | |3   I*\/ 3 |
|- - -------|*|- + -------|
\2      2   / \2      2   /
(32−3i2)(32+3i2)\left(\frac{3}{2} - \frac{\sqrt{3} i}{2}\right) \left(\frac{3}{2} + \frac{\sqrt{3} i}{2}\right) 
=
3
33 
3
Numerical answer [src] 
x1 = 1.5 - 0.866025403784439*i
x2 = 1.5 + 0.866025403784439*i
x2 = 1.5 + 0.866025403784439*i
The graph
x^2-3*x+3=0 equation
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