z^2-4z+1=0

Solution for z^2-4z+1=0 equation:

Simplifying
z2 + -4z + 1 = 0

Reorder the terms:
1 + -4z + z2 = 0

Solving
1 + -4z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -4z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -4z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4z + z2 = 0 + -1
-4z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-4z + z2 = -1

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = -1 + 4

Reorder the terms:
4 + -4z + z2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + -4z + z2 = 3

Factor a perfect square on the left side:
(z + -2)(z + -2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + -2) equal to 1.732050808 and -1.732050808.

Subproblem 1
z + -2 = 1.732050808

Simplifying
z + -2 = 1.732050808

Reorder the terms:
-2 + z = 1.732050808

Solving
-2 + z = 1.732050808

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = 1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + z = 1.732050808 + 2
z = 1.732050808 + 2

Combine like terms: 1.732050808 + 2 = 3.732050808
z = 3.732050808

Simplifying
z = 3.732050808

Subproblem 2
z + -2 = -1.732050808

Simplifying
z + -2 = -1.732050808

Reorder the terms:
-2 + z = -1.732050808

Solving
-2 + z = -1.732050808

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = -1.732050808 + 2

Combine like terms: -2 + 2 = 0
0 + z = -1.732050808 + 2
z = -1.732050808 + 2

Combine like terms: -1.732050808 + 2 = 0.267949192
z = 0.267949192

Simplifying
z = 0.267949192

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.732050808, 0.267949192}