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x^2-x+3=0

x^2-x+3=0 equation

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Numerical solution:

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The solution

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 2            
x  - x + 3 = 0
(x2−x)+3=0\left(x^{2} - 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=−1b = -1
c=3c = 3
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1) * (3) = -11

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=12+11i2x_{1} = \frac{1}{2} + \frac{\sqrt{11} i}{2}
x2=12−11i2x_{2} = \frac{1}{2} - \frac{\sqrt{11} 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=−1p = -1
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=1x_{1} + x_{2} = 1
x1x2=3x_{1} x_{2} = 3
The graph
0123456-5-4-3-2-1020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
        ____           ____
1   I*\/ 11    1   I*\/ 11 
- - -------- + - + --------
2      2       2      2
(12−11i2)+(12+11i2)\left(\frac{1}{2} - \frac{\sqrt{11} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{11} i}{2}\right) 
=
1
11 
product
/        ____\ /        ____\
|1   I*\/ 11 | |1   I*\/ 11 |
|- - --------|*|- + --------|
\2      2    / \2      2    /
(12−11i2)(12+11i2)\left(\frac{1}{2} - \frac{\sqrt{11} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{11} i}{2}\right) 
=
3
33 
3
Rapid solution [src] 
             ____
     1   I*\/ 11 
x1 = - - --------
     2      2
x1=12−11i2x_{1} = \frac{1}{2} - \frac{\sqrt{11} i}{2} 
             ____
     1   I*\/ 11 
x2 = - + --------
     2      2
x2=12+11i2x_{2} = \frac{1}{2} + \frac{\sqrt{11} i}{2} 
x2 = 1/2 + sqrt(11)*i/2
Numerical answer [src] 
x1 = 0.5 + 1.6583123951777*i
x2 = 0.5 - 1.6583123951777*i
x2 = 0.5 - 1.6583123951777*i
The graph
x^2-x+3=0 equation
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