3z^2-6z-6=0

Solution for 3z^2-6z-6=0 equation:

Simplifying
3z2 + -6z + -6 = 0

Reorder the terms:
-6 + -6z + 3z2 = 0

Solving
-6 + -6z + 3z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-2 + -2z + z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-2 + -2z + z2)' equal to zero and attempt to solve:

Simplifying
-2 + -2z + z2 = 0

Solving
-2 + -2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -2z + z2 = 0 + 2

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

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

The z term is -2z.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2z + 1 + z2 = 2 + 1

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

Combine like terms: 2 + 1 = 3
1 + -2z + z2 = 3

Factor a perfect square on the left side:
(z + -1)(z + -1) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + -1) equal to 1.732050808 and -1.732050808.

Subproblem 1
z + -1 = 1.732050808

Simplifying
z + -1 = 1.732050808

Reorder the terms:
-1 + z = 1.732050808

Solving
-1 + z = 1.732050808

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = 1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + z = 1.732050808 + 1
z = 1.732050808 + 1

Combine like terms: 1.732050808 + 1 = 2.732050808
z = 2.732050808

Simplifying
z = 2.732050808

Subproblem 2
z + -1 = -1.732050808

Simplifying
z + -1 = -1.732050808

Reorder the terms:
-1 + z = -1.732050808

Solving
-1 + z = -1.732050808

Solving for variable 'z'.

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

Add '1' to each side of the equation.
-1 + 1 + z = -1.732050808 + 1

Combine like terms: -1 + 1 = 0
0 + z = -1.732050808 + 1
z = -1.732050808 + 1

Combine like terms: -1.732050808 + 1 = -0.732050808
z = -0.732050808

Simplifying
z = -0.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.732050808, -0.732050808}
Solution
z = {2.732050808, -0.732050808}