3z(4z+2)=88

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Solution for 3z(4z+2)=88 equation: 


Simplifying
3z(4z + 2) = 88

Reorder the terms:
3z(2 + 4z) = 88
(2 * 3z + 4z * 3z) = 88
(6z + 12z2) = 88

Solving
6z + 12z2 = 88

Solving for variable 'z'.

Reorder the terms:
-88 + 6z + 12z2 = 88 + -88

Combine like terms: 88 + -88 = 0
-88 + 6z + 12z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-44 + 3z + 6z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-44 + 3z + 6z2)' equal to zero and attempt to solve:

Simplifying
-44 + 3z + 6z2 = 0

Solving
-44 + 3z + 6z2 = 0

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

Divide each side by '6'.
-7.333333333 + 0.5z + z2 = 0

Move the constant term to the right:

Add '7.333333333' to each side of the equation.
-7.333333333 + 0.5z + 7.333333333 + z2 = 0 + 7.333333333

Reorder the terms:
-7.333333333 + 7.333333333 + 0.5z + z2 = 0 + 7.333333333

Combine like terms: -7.333333333 + 7.333333333 = 0.000000000
0.000000000 + 0.5z + z2 = 0 + 7.333333333
0.5z + z2 = 0 + 7.333333333

Combine like terms: 0 + 7.333333333 = 7.333333333
0.5z + z2 = 7.333333333

The z term is 0.5z.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5z + 0.0625 + z2 = 7.333333333 + 0.0625

Reorder the terms:
0.0625 + 0.5z + z2 = 7.333333333 + 0.0625

Combine like terms: 7.333333333 + 0.0625 = 7.395833333
0.0625 + 0.5z + z2 = 7.395833333

Factor a perfect square on the left side:
(z + 0.25)(z + 0.25) = 7.395833333

Calculate the square root of the right side: 2.719528145

Break this problem into two subproblems by setting 
(z + 0.25) equal to 2.719528145 and -2.719528145.


Subproblem 1
z + 0.25 = 2.719528145

Simplifying
z + 0.25 = 2.719528145

Reorder the terms:
0.25 + z = 2.719528145

Solving
0.25 + z = 2.719528145

Solving for variable 'z'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + z = 2.719528145 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + z = 2.719528145 + -0.25
z = 2.719528145 + -0.25

Combine like terms: 2.719528145 + -0.25 = 2.469528145
z = 2.469528145

Simplifying
z = 2.469528145


Subproblem 2
z + 0.25 = -2.719528145

Simplifying
z + 0.25 = -2.719528145

Reorder the terms:
0.25 + z = -2.719528145

Solving
0.25 + z = -2.719528145

Solving for variable 'z'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + z = -2.719528145 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + z = -2.719528145 + -0.25
z = -2.719528145 + -0.25

Combine like terms: -2.719528145 + -0.25 = -2.969528145
z = -2.969528145

Simplifying
z = -2.969528145


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.469528145, -2.969528145}
Solution
z = {2.469528145, -2.969528145}

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