3Z(2Z+7)+12=0

Solution for 3Z(2Z+7)+12=0 equation:

Simplifying
3Z(2Z + 7) + 12 = 0

Reorder the terms:
3Z(7 + 2Z) + 12 = 0
(7 * 3Z + 2Z * 3Z) + 12 = 0
(21Z + 6Z2) + 12 = 0

Reorder the terms:
12 + 21Z + 6Z2 = 0

Solving
12 + 21Z + 6Z2 = 0

Solving for variable 'Z'.

Factor out the Greatest Common Factor (GCF), '3'.
3(4 + 7Z + 2Z2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(4 + 7Z + 2Z2)' equal to zero and attempt to solve:

Simplifying
4 + 7Z + 2Z2 = 0

Solving
4 + 7Z + 2Z2 = 0

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

Divide each side by '2'.
2 + 3.5Z + Z2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 3.5Z + -2 + Z2 = 0 + -2

Reorder the terms:
2 + -2 + 3.5Z + Z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 3.5Z + Z2 = 0 + -2
3.5Z + Z2 = 0 + -2

Combine like terms: 0 + -2 = -2
3.5Z + Z2 = -2

The Z term is 3.5Z.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5Z + 3.0625 + Z2 = -2 + 3.0625

Reorder the terms:
3.0625 + 3.5Z + Z2 = -2 + 3.0625

Combine like terms: -2 + 3.0625 = 1.0625
3.0625 + 3.5Z + Z2 = 1.0625

Factor a perfect square on the left side:
(Z + 1.75)(Z + 1.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(Z + 1.75) equal to 1.030776406 and -1.030776406.

Subproblem 1
Z + 1.75 = 1.030776406

Simplifying
Z + 1.75 = 1.030776406

Reorder the terms:
1.75 + Z = 1.030776406

Solving
1.75 + Z = 1.030776406

Solving for variable 'Z'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + Z = 1.030776406 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + Z = 1.030776406 + -1.75
Z = 1.030776406 + -1.75

Combine like terms: 1.030776406 + -1.75 = -0.719223594
Z = -0.719223594

Simplifying
Z = -0.719223594

Subproblem 2
Z + 1.75 = -1.030776406

Simplifying
Z + 1.75 = -1.030776406

Reorder the terms:
1.75 + Z = -1.030776406

Solving
1.75 + Z = -1.030776406

Solving for variable 'Z'.

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

Add '-1.75' to each side of the equation.
1.75 + -1.75 + Z = -1.030776406 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + Z = -1.030776406 + -1.75
Z = -1.030776406 + -1.75

Combine like terms: -1.030776406 + -1.75 = -2.780776406
Z = -2.780776406

Simplifying
Z = -2.780776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
Z = {-0.719223594, -2.780776406}
Solution
Z = {-0.719223594, -2.780776406}