0=n^2-3n+1

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

Simplifying
0 = n2 + -3n + 1

Reorder the terms:
0 = 1 + -3n + n2

Solving
0 = 1 + -3n + n2

Solving for variable 'n'.

Combine like terms: 0 + -1 = -1
-1 + 3n + -1n2 = 1 + -3n + n2 + -1 + 3n + -1n2

Reorder the terms:
-1 + 3n + -1n2 = 1 + -1 + -3n + 3n + n2 + -1n2

Combine like terms: 1 + -1 = 0
-1 + 3n + -1n2 = 0 + -3n + 3n + n2 + -1n2
-1 + 3n + -1n2 = -3n + 3n + n2 + -1n2

Combine like terms: -3n + 3n = 0
-1 + 3n + -1n2 = 0 + n2 + -1n2
-1 + 3n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-1 + 3n + -1n2 = 0

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

Divide each side by '-1'.
1 + -3n + n2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -3n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3n + n2 = 0 + -1
-3n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3n + n2 = -1

The n term is -3n.  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.
-3n + 2.25 + n2 = -1 + 2.25

Reorder the terms:
2.25 + -3n + n2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + -3n + n2 = 1.25

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

Calculate the square root of the right side: 1.118033989

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

Subproblem 1
n + -1.5 = 1.118033989

Simplifying
n + -1.5 = 1.118033989

Reorder the terms:
-1.5 + n = 1.118033989

Solving
-1.5 + n = 1.118033989

Solving for variable 'n'.

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

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

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

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

Simplifying
n = 2.618033989

Subproblem 2
n + -1.5 = -1.118033989

Simplifying
n + -1.5 = -1.118033989

Reorder the terms:
-1.5 + n = -1.118033989

Solving
-1.5 + n = -1.118033989

Solving for variable 'n'.

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

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

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

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

Simplifying
n = 0.381966011

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