z^2-8z+1=0

Solution for z^2-8z+1=0 equation:

Simplifying
z2 + -8z + 1 = 0

Reorder the terms:
1 + -8z + z2 = 0

Solving
1 + -8z + 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 + -8z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -8z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -8z + z2 = 0 + -1
-8z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-8z + z2 = -1

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = -1 + 16

Reorder the terms:
16 + -8z + z2 = -1 + 16

Combine like terms: -1 + 16 = 15
16 + -8z + z2 = 15

Factor a perfect square on the left side:
(z + -4)(z + -4) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(z + -4) equal to 3.872983346 and -3.872983346.

Subproblem 1
z + -4 = 3.872983346

Simplifying
z + -4 = 3.872983346

Reorder the terms:
-4 + z = 3.872983346

Solving
-4 + z = 3.872983346

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = 3.872983346 + 4

Combine like terms: -4 + 4 = 0
0 + z = 3.872983346 + 4
z = 3.872983346 + 4

Combine like terms: 3.872983346 + 4 = 7.872983346
z = 7.872983346

Simplifying
z = 7.872983346

Subproblem 2
z + -4 = -3.872983346

Simplifying
z + -4 = -3.872983346

Reorder the terms:
-4 + z = -3.872983346

Solving
-4 + z = -3.872983346

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = -3.872983346 + 4

Combine like terms: -4 + 4 = 0
0 + z = -3.872983346 + 4
z = -3.872983346 + 4

Combine like terms: -3.872983346 + 4 = 0.127016654
z = 0.127016654

Simplifying
z = 0.127016654

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {7.872983346, 0.127016654}