z^2-6z+[1]=0

Solution for z^2-6z+[1]=0 equation:

Simplifying
z2 + -6z + [1] = 0
z2 + -6z + 1 = 0

Reorder the terms:
1 + -6z + z2 = 0

Solving
1 + -6z + 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 + -6z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -6z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -6z + z2 = 0 + -1
-6z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-6z + z2 = -1

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = -1 + 9

Reorder the terms:
9 + -6z + z2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + -6z + z2 = 8

Factor a perfect square on the left side:
(z + -3)(z + -3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(z + -3) equal to 2.828427125 and -2.828427125.

Subproblem 1
z + -3 = 2.828427125

Simplifying
z + -3 = 2.828427125

Reorder the terms:
-3 + z = 2.828427125

Solving
-3 + z = 2.828427125

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = 2.828427125 + 3

Combine like terms: -3 + 3 = 0
0 + z = 2.828427125 + 3
z = 2.828427125 + 3

Combine like terms: 2.828427125 + 3 = 5.828427125
z = 5.828427125

Simplifying
z = 5.828427125

Subproblem 2
z + -3 = -2.828427125

Simplifying
z + -3 = -2.828427125

Reorder the terms:
-3 + z = -2.828427125

Solving
-3 + z = -2.828427125

Solving for variable 'z'.

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

Add '3' to each side of the equation.
-3 + 3 + z = -2.828427125 + 3

Combine like terms: -3 + 3 = 0
0 + z = -2.828427125 + 3
z = -2.828427125 + 3

Combine like terms: -2.828427125 + 3 = 0.171572875
z = 0.171572875

Simplifying
z = 0.171572875

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {5.828427125, 0.171572875}