3(2z+25)2(z-1)=78

Solution for 3(2z+25)2(z-1)=78 equation:

Simplifying
3(2z + 25) * 2(z + -1) = 78

Reorder the terms:
3(25 + 2z) * 2(z + -1) = 78

Reorder the terms:
3(25 + 2z) * 2(-1 + z) = 78

Reorder the terms for easier multiplication:
3 * 2(25 + 2z)(-1 + z) = 78

Multiply 3 * 2
6(25 + 2z)(-1 + z) = 78

Multiply (25 + 2z) * (-1 + z)
6(25(-1 + z) + 2z * (-1 + z)) = 78
6((-1 * 25 + z * 25) + 2z * (-1 + z)) = 78
6((-25 + 25z) + 2z * (-1 + z)) = 78
6(-25 + 25z + (-1 * 2z + z * 2z)) = 78
6(-25 + 25z + (-2z + 2z2)) = 78

Combine like terms: 25z + -2z = 23z
6(-25 + 23z + 2z2) = 78
(-25 * 6 + 23z * 6 + 2z2 * 6) = 78
(-150 + 138z + 12z2) = 78

Solving
-150 + 138z + 12z2 = 78

Solving for variable 'z'.

Reorder the terms:
-150 + -78 + 138z + 12z2 = 78 + -78

Combine like terms: -150 + -78 = -228
-228 + 138z + 12z2 = 78 + -78

Combine like terms: 78 + -78 = 0
-228 + 138z + 12z2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-38 + 23z + 2z2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(-38 + 23z + 2z2)' equal to zero and attempt to solve:

Simplifying
-38 + 23z + 2z2 = 0

Solving
-38 + 23z + 2z2 = 0

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

Divide each side by '2'.
-19 + 11.5z + z2 = 0

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 11.5z + 19 + z2 = 0 + 19

Reorder the terms:
-19 + 19 + 11.5z + z2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 11.5z + z2 = 0 + 19
11.5z + z2 = 0 + 19

Combine like terms: 0 + 19 = 19
11.5z + z2 = 19

The z term is 11.5z.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5z + 33.0625 + z2 = 19 + 33.0625

Reorder the terms:
33.0625 + 11.5z + z2 = 19 + 33.0625

Combine like terms: 19 + 33.0625 = 52.0625
33.0625 + 11.5z + z2 = 52.0625

Factor a perfect square on the left side:
(z + 5.75)(z + 5.75) = 52.0625

Calculate the square root of the right side: 7.215434845

Break this problem into two subproblems by setting 
(z + 5.75) equal to 7.215434845 and -7.215434845.

Subproblem 1
z + 5.75 = 7.215434845

Simplifying
z + 5.75 = 7.215434845

Reorder the terms:
5.75 + z = 7.215434845

Solving
5.75 + z = 7.215434845

Solving for variable 'z'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + z = 7.215434845 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + z = 7.215434845 + -5.75
z = 7.215434845 + -5.75

Combine like terms: 7.215434845 + -5.75 = 1.465434845
z = 1.465434845

Simplifying
z = 1.465434845

Subproblem 2
z + 5.75 = -7.215434845

Simplifying
z + 5.75 = -7.215434845

Reorder the terms:
5.75 + z = -7.215434845

Solving
5.75 + z = -7.215434845

Solving for variable 'z'.

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

Add '-5.75' to each side of the equation.
5.75 + -5.75 + z = -7.215434845 + -5.75

Combine like terms: 5.75 + -5.75 = 0.00
0.00 + z = -7.215434845 + -5.75
z = -7.215434845 + -5.75

Combine like terms: -7.215434845 + -5.75 = -12.965434845
z = -12.965434845

Simplifying
z = -12.965434845

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.465434845, -12.965434845}
Solution
z = {1.465434845, -12.965434845}