0=x^2-3x+1

Solution for 0=x^2-3x+1 equation:

Simplifying
0 = x2 + -3x + 1

Reorder the terms:
0 = 1 + -3x + x2

Solving
0 = 1 + -3x + x2

Solving for variable 'x'.

Combine like terms: 0 + -1 = -1
-1 + 3x + -1x2 = 1 + -3x + x2 + -1 + 3x + -1x2

Reorder the terms:
-1 + 3x + -1x2 = 1 + -1 + -3x + 3x + x2 + -1x2

Combine like terms: 1 + -1 = 0
-1 + 3x + -1x2 = 0 + -3x + 3x + x2 + -1x2
-1 + 3x + -1x2 = -3x + 3x + x2 + -1x2

Combine like terms: -3x + 3x = 0
-1 + 3x + -1x2 = 0 + x2 + -1x2
-1 + 3x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-1 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
1 + -3x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -3x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3x + x2 = 0 + -1
-3x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3x + x2 = -1

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = -1 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3x + x2 = 1.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
x + -1.5 = 1.118033989

Simplifying
x + -1.5 = 1.118033989

Reorder the terms:
-1.5 + x = 1.118033989

Solving
-1.5 + x = 1.118033989

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 1.118033989 + 1.5
x = 1.118033989 + 1.5

Combine like terms: 1.118033989 + 1.5 = 2.618033989
x = 2.618033989

Simplifying
x = 2.618033989

Subproblem 2
x + -1.5 = -1.118033989

Simplifying
x + -1.5 = -1.118033989

Reorder the terms:
-1.5 + x = -1.118033989

Solving
-1.5 + x = -1.118033989

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -1.118033989 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -1.118033989 + 1.5
x = -1.118033989 + 1.5

Combine like terms: -1.118033989 + 1.5 = 0.381966011
x = 0.381966011

Simplifying
x = 0.381966011

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.618033989, 0.381966011}