3(6z+7)10z+45=180

Solution for 3(6z+7)10z+45=180 equation:

Simplifying
3(6z + 7) * 10z + 45 = 180

Reorder the terms:
3(7 + 6z) * 10z + 45 = 180

Reorder the terms for easier multiplication:
3 * 10z(7 + 6z) + 45 = 180

Multiply 3 * 10
30z(7 + 6z) + 45 = 180
(7 * 30z + 6z * 30z) + 45 = 180
(210z + 180z2) + 45 = 180

Reorder the terms:
45 + 210z + 180z2 = 180

Solving
45 + 210z + 180z2 = 180

Solving for variable 'z'.

Reorder the terms:
45 + -180 + 210z + 180z2 = 180 + -180

Combine like terms: 45 + -180 = -135
-135 + 210z + 180z2 = 180 + -180

Combine like terms: 180 + -180 = 0
-135 + 210z + 180z2 = 0

Factor out the Greatest Common Factor (GCF), '15'.
15(-9 + 14z + 12z2) = 0

Ignore the factor 15.
Subproblem 1
Set the factor '(-9 + 14z + 12z2)' equal to zero and attempt to solve:

Simplifying
-9 + 14z + 12z2 = 0

Solving
-9 + 14z + 12z2 = 0

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

Divide each side by '12'.
-0.75 + 1.166666667z + z2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.166666667z + 0.75 + z2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.166666667z + z2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.166666667z + z2 = 0 + 0.75
1.166666667z + z2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.166666667z + z2 = 0.75

The z term is 1.166666667z.  Take half its coefficient (0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
1.166666667z + 0.3402777780 + z2 = 0.75 + 0.3402777780

Reorder the terms:
0.3402777780 + 1.166666667z + z2 = 0.75 + 0.3402777780

Combine like terms: 0.75 + 0.3402777780 = 1.090277778
0.3402777780 + 1.166666667z + z2 = 1.090277778

Factor a perfect square on the left side:
(z + 0.5833333335)(z + 0.5833333335) = 1.090277778

Calculate the square root of the right side: 1.044163674

Break this problem into two subproblems by setting 
(z + 0.5833333335) equal to 1.044163674 and -1.044163674.

Subproblem 1
z + 0.5833333335 = 1.044163674

Simplifying
z + 0.5833333335 = 1.044163674

Reorder the terms:
0.5833333335 + z = 1.044163674

Solving
0.5833333335 + z = 1.044163674

Solving for variable 'z'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + z = 1.044163674 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + z = 1.044163674 + -0.5833333335
z = 1.044163674 + -0.5833333335

Combine like terms: 1.044163674 + -0.5833333335 = 0.4608303405
z = 0.4608303405

Simplifying
z = 0.4608303405

Subproblem 2
z + 0.5833333335 = -1.044163674

Simplifying
z + 0.5833333335 = -1.044163674

Reorder the terms:
0.5833333335 + z = -1.044163674

Solving
0.5833333335 + z = -1.044163674

Solving for variable 'z'.

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

Add '-0.5833333335' to each side of the equation.
0.5833333335 + -0.5833333335 + z = -1.044163674 + -0.5833333335

Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000
0.0000000000 + z = -1.044163674 + -0.5833333335
z = -1.044163674 + -0.5833333335

Combine like terms: -1.044163674 + -0.5833333335 = -1.6274970075
z = -1.6274970075

Simplifying
z = -1.6274970075

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.4608303405, -1.6274970075}
Solution
z = {0.4608303405, -1.6274970075}