3z^2+6z-18=0

Solution for 3z^2+6z-18=0 equation:

Simplifying
3z2 + 6z + -18 = 0

Reorder the terms:
-18 + 6z + 3z2 = 0

Solving
-18 + 6z + 3z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-6 + 2z + z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-6 + 2z + z2)' equal to zero and attempt to solve:

Simplifying
-6 + 2z + z2 = 0

Solving
-6 + 2z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 2z + 6 + z2 = 0 + 6

Reorder the terms:
-6 + 6 + 2z + z2 = 0 + 6

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

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

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 = 6 + 1

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

Combine like terms: 6 + 1 = 7
1 + 2z + z2 = 7

Factor a perfect square on the left side:
(z + 1)(z + 1) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(z + 1) equal to 2.645751311 and -2.645751311.

Subproblem 1
z + 1 = 2.645751311

Simplifying
z + 1 = 2.645751311

Reorder the terms:
1 + z = 2.645751311

Solving
1 + z = 2.645751311

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 = 2.645751311 + -1

Combine like terms: 1 + -1 = 0
0 + z = 2.645751311 + -1
z = 2.645751311 + -1

Combine like terms: 2.645751311 + -1 = 1.645751311
z = 1.645751311

Simplifying
z = 1.645751311

Subproblem 2
z + 1 = -2.645751311

Simplifying
z + 1 = -2.645751311

Reorder the terms:
1 + z = -2.645751311

Solving
1 + z = -2.645751311

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 = -2.645751311 + -1

Combine like terms: 1 + -1 = 0
0 + z = -2.645751311 + -1
z = -2.645751311 + -1

Combine like terms: -2.645751311 + -1 = -3.645751311
z = -3.645751311

Simplifying
z = -3.645751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.645751311, -3.645751311}
Solution
z = {1.645751311, -3.645751311}