6z^2+32z+18=0

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Solution for 6z^2+32z+18=0 equation: 


Simplifying
6z2 + 32z + 18 = 0

Reorder the terms:
18 + 32z + 6z2 = 0

Solving
18 + 32z + 6z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '2'.
2(9 + 16z + 3z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(9 + 16z + 3z2)' equal to zero and attempt to solve:

Simplifying
9 + 16z + 3z2 = 0

Solving
9 + 16z + 3z2 = 0

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

Divide each side by '3'.
3 + 5.333333333z + z2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5.333333333z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 5.333333333z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5.333333333z + z2 = 0 + -3
5.333333333z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
5.333333333z + z2 = -3

The z term is 5.333333333z.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333z + 7.111111113 + z2 = -3 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333z + z2 = -3 + 7.111111113

Combine like terms: -3 + 7.111111113 = 4.111111113
7.111111113 + 5.333333333z + z2 = 4.111111113

Factor a perfect square on the left side:
(z + 2.666666667)(z + 2.666666667) = 4.111111113

Calculate the square root of the right side: 2.027587511

Break this problem into two subproblems by setting 
(z + 2.666666667) equal to 2.027587511 and -2.027587511.


Subproblem 1
z + 2.666666667 = 2.027587511

Simplifying
z + 2.666666667 = 2.027587511

Reorder the terms:
2.666666667 + z = 2.027587511

Solving
2.666666667 + z = 2.027587511

Solving for variable 'z'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + z = 2.027587511 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + z = 2.027587511 + -2.666666667
z = 2.027587511 + -2.666666667

Combine like terms: 2.027587511 + -2.666666667 = -0.639079156
z = -0.639079156

Simplifying
z = -0.639079156


Subproblem 2
z + 2.666666667 = -2.027587511

Simplifying
z + 2.666666667 = -2.027587511

Reorder the terms:
2.666666667 + z = -2.027587511

Solving
2.666666667 + z = -2.027587511

Solving for variable 'z'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + z = -2.027587511 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + z = -2.027587511 + -2.666666667
z = -2.027587511 + -2.666666667

Combine like terms: -2.027587511 + -2.666666667 = -4.694254178
z = -4.694254178

Simplifying
z = -4.694254178


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.639079156, -4.694254178}
Solution
z = {-0.639079156, -4.694254178}

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