12z(4z+6)=82

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Solution for 12z(4z+6)=82 equation: 


Simplifying
12z(4z + 6) = 82

Reorder the terms:
12z(6 + 4z) = 82
(6 * 12z + 4z * 12z) = 82
(72z + 48z2) = 82

Solving
72z + 48z2 = 82

Solving for variable 'z'.

Reorder the terms:
-82 + 72z + 48z2 = 82 + -82

Combine like terms: 82 + -82 = 0
-82 + 72z + 48z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-41 + 36z + 24z2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-41 + 36z + 24z2)' equal to zero and attempt to solve:

Simplifying
-41 + 36z + 24z2 = 0

Solving
-41 + 36z + 24z2 = 0

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

Divide each side by '24'.
-1.708333333 + 1.5z + z2 = 0

Move the constant term to the right:

Add '1.708333333' to each side of the equation.
-1.708333333 + 1.5z + 1.708333333 + z2 = 0 + 1.708333333

Reorder the terms:
-1.708333333 + 1.708333333 + 1.5z + z2 = 0 + 1.708333333

Combine like terms: -1.708333333 + 1.708333333 = 0.000000000
0.000000000 + 1.5z + z2 = 0 + 1.708333333
1.5z + z2 = 0 + 1.708333333

Combine like terms: 0 + 1.708333333 = 1.708333333
1.5z + z2 = 1.708333333

The z term is 1.5z.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5z + 0.5625 + z2 = 1.708333333 + 0.5625

Reorder the terms:
0.5625 + 1.5z + z2 = 1.708333333 + 0.5625

Combine like terms: 1.708333333 + 0.5625 = 2.270833333
0.5625 + 1.5z + z2 = 2.270833333

Factor a perfect square on the left side:
(z + 0.75)(z + 0.75) = 2.270833333

Calculate the square root of the right side: 1.506928443

Break this problem into two subproblems by setting 
(z + 0.75) equal to 1.506928443 and -1.506928443.


Subproblem 1
z + 0.75 = 1.506928443

Simplifying
z + 0.75 = 1.506928443

Reorder the terms:
0.75 + z = 1.506928443

Solving
0.75 + z = 1.506928443

Solving for variable 'z'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + z = 1.506928443 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + z = 1.506928443 + -0.75
z = 1.506928443 + -0.75

Combine like terms: 1.506928443 + -0.75 = 0.756928443
z = 0.756928443

Simplifying
z = 0.756928443


Subproblem 2
z + 0.75 = -1.506928443

Simplifying
z + 0.75 = -1.506928443

Reorder the terms:
0.75 + z = -1.506928443

Solving
0.75 + z = -1.506928443

Solving for variable 'z'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + z = -1.506928443 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + z = -1.506928443 + -0.75
z = -1.506928443 + -0.75

Combine like terms: -1.506928443 + -0.75 = -2.256928443
z = -2.256928443

Simplifying
z = -2.256928443


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.756928443, -2.256928443}
Solution
z = {0.756928443, -2.256928443}

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