2z^2-6z=1

Solution for 2z^2-6z=1 equation:

Simplifying
2z2 + -6z = 1

Reorder the terms:
-6z + 2z2 = 1

Solving
-6z + 2z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + -6z + 2z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -6z + 2z2 = 0

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

Divide each side by '2'.
-0.5 + -3z + z2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -3z + 0.5 + z2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -3z + z2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -3z + z2 = 0 + 0.5
-3z + z2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-3z + z2 = 0.5

The z term is -3z.  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.
-3z + 2.25 + z2 = 0.5 + 2.25

Reorder the terms:
2.25 + -3z + z2 = 0.5 + 2.25

Combine like terms: 0.5 + 2.25 = 2.75
2.25 + -3z + z2 = 2.75

Factor a perfect square on the left side:
(z + -1.5)(z + -1.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(z + -1.5) equal to 1.658312395 and -1.658312395.

Subproblem 1
z + -1.5 = 1.658312395

Simplifying
z + -1.5 = 1.658312395

Reorder the terms:
-1.5 + z = 1.658312395

Solving
-1.5 + z = 1.658312395

Solving for variable 'z'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = 1.658312395 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = 1.658312395 + 1.5
z = 1.658312395 + 1.5

Combine like terms: 1.658312395 + 1.5 = 3.158312395
z = 3.158312395

Simplifying
z = 3.158312395

Subproblem 2
z + -1.5 = -1.658312395

Simplifying
z + -1.5 = -1.658312395

Reorder the terms:
-1.5 + z = -1.658312395

Solving
-1.5 + z = -1.658312395

Solving for variable 'z'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + z = -1.658312395 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + z = -1.658312395 + 1.5
z = -1.658312395 + 1.5

Combine like terms: -1.658312395 + 1.5 = -0.158312395
z = -0.158312395

Simplifying
z = -0.158312395

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.158312395, -0.158312395}