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x^2-3*x+1=0

x^2-3*x+1=0 equation

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Numerical solution:

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The solution

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

(-3)^2 - 4 * (1) * (1) = 5

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=52+32x_{1} = \frac{\sqrt{5}}{2} + \frac{3}{2}
x2=32−52x_{2} = \frac{3}{2} - \frac{\sqrt{5}}{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=1q = 1
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=3x_{1} + x_{2} = 3
x1x2=1x_{1} x_{2} = 1
The graph
05-10-51015200-100
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           ___
     3   \/ 5 
x1 = - - -----
     2     2
x1=32−52x_{1} = \frac{3}{2} - \frac{\sqrt{5}}{2} 
           ___
     3   \/ 5 
x2 = - + -----
     2     2
x2=52+32x_{2} = \frac{\sqrt{5}}{2} + \frac{3}{2} 
x2 = sqrt(5)/2 + 3/2
Sum and product of roots [src] 
sum
      ___         ___
3   \/ 5    3   \/ 5 
- - ----- + - + -----
2     2     2     2
(32−52)+(52+32)\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right) + \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right) 
=
3
33 
product
/      ___\ /      ___\
|3   \/ 5 | |3   \/ 5 |
|- - -----|*|- + -----|
\2     2  / \2     2  /
(32−52)(52+32)\left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right) \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right) 
=
1
11 
1
Numerical answer [src] 
x1 = 2.61803398874989
x2 = 0.381966011250105
x2 = 0.381966011250105
The graph
x^2-3*x+1=0 equation
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