-8z(z+3)+5(3z-3)=6(z-6)+8

Solution for -8z(z+3)+5(3z-3)=6(z-6)+8 equation:

Simplifying
-8z(z + 3) + 5(3z + -3) = 6(z + -6) + 8

Reorder the terms:
-8z(3 + z) + 5(3z + -3) = 6(z + -6) + 8
(3 * -8z + z * -8z) + 5(3z + -3) = 6(z + -6) + 8
(-24z + -8z2) + 5(3z + -3) = 6(z + -6) + 8

Reorder the terms:
-24z + -8z2 + 5(-3 + 3z) = 6(z + -6) + 8
-24z + -8z2 + (-3 * 5 + 3z * 5) = 6(z + -6) + 8
-24z + -8z2 + (-15 + 15z) = 6(z + -6) + 8

Reorder the terms:
-15 + -24z + 15z + -8z2 = 6(z + -6) + 8

Combine like terms: -24z + 15z = -9z
-15 + -9z + -8z2 = 6(z + -6) + 8

Reorder the terms:
-15 + -9z + -8z2 = 6(-6 + z) + 8
-15 + -9z + -8z2 = (-6 * 6 + z * 6) + 8
-15 + -9z + -8z2 = (-36 + 6z) + 8

Reorder the terms:
-15 + -9z + -8z2 = -36 + 8 + 6z

Combine like terms: -36 + 8 = -28
-15 + -9z + -8z2 = -28 + 6z

Solving
-15 + -9z + -8z2 = -28 + 6z

Solving for variable 'z'.

Reorder the terms:
-15 + 28 + -9z + -6z + -8z2 = -28 + 6z + 28 + -6z

Combine like terms: -15 + 28 = 13
13 + -9z + -6z + -8z2 = -28 + 6z + 28 + -6z

Combine like terms: -9z + -6z = -15z
13 + -15z + -8z2 = -28 + 6z + 28 + -6z

Reorder the terms:
13 + -15z + -8z2 = -28 + 28 + 6z + -6z

Combine like terms: -28 + 28 = 0
13 + -15z + -8z2 = 0 + 6z + -6z
13 + -15z + -8z2 = 6z + -6z

Combine like terms: 6z + -6z = 0
13 + -15z + -8z2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-1.625 + 1.875z + z2 = 0

Move the constant term to the right:

Add '1.625' to each side of the equation.
-1.625 + 1.875z + 1.625 + z2 = 0 + 1.625

Reorder the terms:
-1.625 + 1.625 + 1.875z + z2 = 0 + 1.625

Combine like terms: -1.625 + 1.625 = 0.000
0.000 + 1.875z + z2 = 0 + 1.625
1.875z + z2 = 0 + 1.625

Combine like terms: 0 + 1.625 = 1.625
1.875z + z2 = 1.625

The z term is 1.875z.  Take half its coefficient (0.9375).
Square it (0.87890625) and add it to both sides.

Add '0.87890625' to each side of the equation.
1.875z + 0.87890625 + z2 = 1.625 + 0.87890625

Reorder the terms:
0.87890625 + 1.875z + z2 = 1.625 + 0.87890625

Combine like terms: 1.625 + 0.87890625 = 2.50390625
0.87890625 + 1.875z + z2 = 2.50390625

Factor a perfect square on the left side:
(z + 0.9375)(z + 0.9375) = 2.50390625

Calculate the square root of the right side: 1.582373613

Break this problem into two subproblems by setting 
(z + 0.9375) equal to 1.582373613 and -1.582373613.

Subproblem 1
z + 0.9375 = 1.582373613

Simplifying
z + 0.9375 = 1.582373613

Reorder the terms:
0.9375 + z = 1.582373613

Solving
0.9375 + z = 1.582373613

Solving for variable 'z'.

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

Add '-0.9375' to each side of the equation.
0.9375 + -0.9375 + z = 1.582373613 + -0.9375

Combine like terms: 0.9375 + -0.9375 = 0.0000
0.0000 + z = 1.582373613 + -0.9375
z = 1.582373613 + -0.9375

Combine like terms: 1.582373613 + -0.9375 = 0.644873613
z = 0.644873613

Simplifying
z = 0.644873613

Subproblem 2
z + 0.9375 = -1.582373613

Simplifying
z + 0.9375 = -1.582373613

Reorder the terms:
0.9375 + z = -1.582373613

Solving
0.9375 + z = -1.582373613

Solving for variable 'z'.

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

Add '-0.9375' to each side of the equation.
0.9375 + -0.9375 + z = -1.582373613 + -0.9375

Combine like terms: 0.9375 + -0.9375 = 0.0000
0.0000 + z = -1.582373613 + -0.9375
z = -1.582373613 + -0.9375

Combine like terms: -1.582373613 + -0.9375 = -2.519873613
z = -2.519873613

Simplifying
z = -2.519873613

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.644873613, -2.519873613}