-2(3-4z)+7z=12z(z+4)

Solution for -2(3-4z)+7z=12z(z+4) equation:

Simplifying
-2(3 + -4z) + 7z = 12z(z + 4)
(3 * -2 + -4z * -2) + 7z = 12z(z + 4)
(-6 + 8z) + 7z = 12z(z + 4)

Combine like terms: 8z + 7z = 15z
-6 + 15z = 12z(z + 4)

Reorder the terms:
-6 + 15z = 12z(4 + z)
-6 + 15z = (4 * 12z + z * 12z)
-6 + 15z = (48z + 12z2)

Solving
-6 + 15z = 48z + 12z2

Solving for variable 'z'.

Combine like terms: 15z + -48z = -33z
-6 + -33z + -12z2 = 48z + 12z2 + -48z + -12z2

Reorder the terms:
-6 + -33z + -12z2 = 48z + -48z + 12z2 + -12z2

Combine like terms: 48z + -48z = 0
-6 + -33z + -12z2 = 0 + 12z2 + -12z2
-6 + -33z + -12z2 = 12z2 + -12z2

Combine like terms: 12z2 + -12z2 = 0
-6 + -33z + -12z2 = 0

Factor out the Greatest Common Factor (GCF), '-3'.
-3(2 + 11z + 4z2) = 0

Ignore the factor -3.
Subproblem 1
Set the factor '(2 + 11z + 4z2)' equal to zero and attempt to solve:

Simplifying
2 + 11z + 4z2 = 0

Solving
2 + 11z + 4z2 = 0

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

Divide each side by '4'.
0.5 + 2.75z + z2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.75z + -0.5 + z2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.75z + z2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.75z + z2 = 0 + -0.5
2.75z + z2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.75z + z2 = -0.5

The z term is 2.75z.  Take half its coefficient (1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
2.75z + 1.890625 + z2 = -0.5 + 1.890625

Reorder the terms:
1.890625 + 2.75z + z2 = -0.5 + 1.890625

Combine like terms: -0.5 + 1.890625 = 1.390625
1.890625 + 2.75z + z2 = 1.390625

Factor a perfect square on the left side:
(z + 1.375)(z + 1.375) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(z + 1.375) equal to 1.179247642 and -1.179247642.

Subproblem 1
z + 1.375 = 1.179247642

Simplifying
z + 1.375 = 1.179247642

Reorder the terms:
1.375 + z = 1.179247642

Solving
1.375 + z = 1.179247642

Solving for variable 'z'.

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

Add '-1.375' to each side of the equation.
1.375 + -1.375 + z = 1.179247642 + -1.375

Combine like terms: 1.375 + -1.375 = 0.000
0.000 + z = 1.179247642 + -1.375
z = 1.179247642 + -1.375

Combine like terms: 1.179247642 + -1.375 = -0.195752358
z = -0.195752358

Simplifying
z = -0.195752358

Subproblem 2
z + 1.375 = -1.179247642

Simplifying
z + 1.375 = -1.179247642

Reorder the terms:
1.375 + z = -1.179247642

Solving
1.375 + z = -1.179247642

Solving for variable 'z'.

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

Add '-1.375' to each side of the equation.
1.375 + -1.375 + z = -1.179247642 + -1.375

Combine like terms: 1.375 + -1.375 = 0.000
0.000 + z = -1.179247642 + -1.375
z = -1.179247642 + -1.375

Combine like terms: -1.179247642 + -1.375 = -2.554247642
z = -2.554247642

Simplifying
z = -2.554247642

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.195752358, -2.554247642}
Solution
z = {-0.195752358, -2.554247642}