3(6z-1)2(z+3)=14(z+1)

Solution for 3(6z-1)2(z+3)=14(z+1) equation:

Simplifying
3(6z + -1) * 2(z + 3) = 14(z + 1)

Reorder the terms:
3(-1 + 6z) * 2(z + 3) = 14(z + 1)

Reorder the terms:
3(-1 + 6z) * 2(3 + z) = 14(z + 1)

Reorder the terms for easier multiplication:
3 * 2(-1 + 6z)(3 + z) = 14(z + 1)

Multiply 3 * 2
6(-1 + 6z)(3 + z) = 14(z + 1)

Multiply (-1 + 6z) * (3 + z)
6(-1(3 + z) + 6z * (3 + z)) = 14(z + 1)
6((3 * -1 + z * -1) + 6z * (3 + z)) = 14(z + 1)
6((-3 + -1z) + 6z * (3 + z)) = 14(z + 1)
6(-3 + -1z + (3 * 6z + z * 6z)) = 14(z + 1)
6(-3 + -1z + (18z + 6z2)) = 14(z + 1)

Combine like terms: -1z + 18z = 17z
6(-3 + 17z + 6z2) = 14(z + 1)
(-3 * 6 + 17z * 6 + 6z2 * 6) = 14(z + 1)
(-18 + 102z + 36z2) = 14(z + 1)

Reorder the terms:
-18 + 102z + 36z2 = 14(1 + z)
-18 + 102z + 36z2 = (1 * 14 + z * 14)
-18 + 102z + 36z2 = (14 + 14z)

Solving
-18 + 102z + 36z2 = 14 + 14z

Solving for variable 'z'.

Reorder the terms:
-18 + -14 + 102z + -14z + 36z2 = 14 + 14z + -14 + -14z

Combine like terms: -18 + -14 = -32
-32 + 102z + -14z + 36z2 = 14 + 14z + -14 + -14z

Combine like terms: 102z + -14z = 88z
-32 + 88z + 36z2 = 14 + 14z + -14 + -14z

Reorder the terms:
-32 + 88z + 36z2 = 14 + -14 + 14z + -14z

Combine like terms: 14 + -14 = 0
-32 + 88z + 36z2 = 0 + 14z + -14z
-32 + 88z + 36z2 = 14z + -14z

Combine like terms: 14z + -14z = 0
-32 + 88z + 36z2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-8 + 22z + 9z2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-8 + 22z + 9z2)' equal to zero and attempt to solve:

Simplifying
-8 + 22z + 9z2 = 0

Solving
-8 + 22z + 9z2 = 0

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

Divide each side by '9'.
-0.8888888889 + 2.444444444z + z2 = 0

Move the constant term to the right:

Add '0.8888888889' to each side of the equation.
-0.8888888889 + 2.444444444z + 0.8888888889 + z2 = 0 + 0.8888888889

Reorder the terms:
-0.8888888889 + 0.8888888889 + 2.444444444z + z2 = 0 + 0.8888888889

Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000
0.0000000000 + 2.444444444z + z2 = 0 + 0.8888888889
2.444444444z + z2 = 0 + 0.8888888889

Combine like terms: 0 + 0.8888888889 = 0.8888888889
2.444444444z + z2 = 0.8888888889

The z term is 2.444444444z.  Take half its coefficient (1.222222222).
Square it (1.493827160) and add it to both sides.

Add '1.493827160' to each side of the equation.
2.444444444z + 1.493827160 + z2 = 0.8888888889 + 1.493827160

Reorder the terms:
1.493827160 + 2.444444444z + z2 = 0.8888888889 + 1.493827160

Combine like terms: 0.8888888889 + 1.493827160 = 2.3827160489
1.493827160 + 2.444444444z + z2 = 2.3827160489

Factor a perfect square on the left side:
(z + 1.222222222)(z + 1.222222222) = 2.3827160489

Calculate the square root of the right side: 1.543604888

Break this problem into two subproblems by setting 
(z + 1.222222222) equal to 1.543604888 and -1.543604888.

Subproblem 1
z + 1.222222222 = 1.543604888

Simplifying
z + 1.222222222 = 1.543604888

Reorder the terms:
1.222222222 + z = 1.543604888

Solving
1.222222222 + z = 1.543604888

Solving for variable 'z'.

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

Add '-1.222222222' to each side of the equation.
1.222222222 + -1.222222222 + z = 1.543604888 + -1.222222222

Combine like terms: 1.222222222 + -1.222222222 = 0.000000000
0.000000000 + z = 1.543604888 + -1.222222222
z = 1.543604888 + -1.222222222

Combine like terms: 1.543604888 + -1.222222222 = 0.321382666
z = 0.321382666

Simplifying
z = 0.321382666

Subproblem 2
z + 1.222222222 = -1.543604888

Simplifying
z + 1.222222222 = -1.543604888

Reorder the terms:
1.222222222 + z = -1.543604888

Solving
1.222222222 + z = -1.543604888

Solving for variable 'z'.

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

Add '-1.222222222' to each side of the equation.
1.222222222 + -1.222222222 + z = -1.543604888 + -1.222222222

Combine like terms: 1.222222222 + -1.222222222 = 0.000000000
0.000000000 + z = -1.543604888 + -1.222222222
z = -1.543604888 + -1.222222222

Combine like terms: -1.543604888 + -1.222222222 = -2.76582711
z = -2.76582711

Simplifying
z = -2.76582711

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.321382666, -2.76582711}
Solution
z = {0.321382666, -2.76582711}