z^2+6z+6=0

Solution for z^2+6z+6=0 equation:

Simplifying
z2 + 6z + 6 = 0

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

Solving
6 + 6z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 6z + -6 + z2 = 0 + -6

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

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

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

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 = -6 + 9

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

Combine like terms: -6 + 9 = 3
9 + 6z + z2 = 3

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

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(z + 3) equal to 1.732050808 and -1.732050808.

Subproblem 1
z + 3 = 1.732050808

Simplifying
z + 3 = 1.732050808

Reorder the terms:
3 + z = 1.732050808

Solving
3 + z = 1.732050808

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 = 1.732050808 + -3

Combine like terms: 3 + -3 = 0
0 + z = 1.732050808 + -3
z = 1.732050808 + -3

Combine like terms: 1.732050808 + -3 = -1.267949192
z = -1.267949192

Simplifying
z = -1.267949192

Subproblem 2
z + 3 = -1.732050808

Simplifying
z + 3 = -1.732050808

Reorder the terms:
3 + z = -1.732050808

Solving
3 + z = -1.732050808

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 = -1.732050808 + -3

Combine like terms: 3 + -3 = 0
0 + z = -1.732050808 + -3
z = -1.732050808 + -3

Combine like terms: -1.732050808 + -3 = -4.732050808
z = -4.732050808

Simplifying
z = -4.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-1.267949192, -4.732050808}