8z+2=2(z-5)z

Solution for 8z+2=2(z-5)z equation:

Simplifying
8z + 2 = 2(z + -5) * z

Reorder the terms:
2 + 8z = 2(z + -5) * z

Reorder the terms:
2 + 8z = 2(-5 + z) * z

Reorder the terms for easier multiplication:
2 + 8z = 2z(-5 + z)
2 + 8z = (-5 * 2z + z * 2z)
2 + 8z = (-10z + 2z2)

Solving
2 + 8z = -10z + 2z2

Solving for variable 'z'.

Combine like terms: 8z + 10z = 18z
2 + 18z + -2z2 = -10z + 2z2 + 10z + -2z2

Reorder the terms:
2 + 18z + -2z2 = -10z + 10z + 2z2 + -2z2

Combine like terms: -10z + 10z = 0
2 + 18z + -2z2 = 0 + 2z2 + -2z2
2 + 18z + -2z2 = 2z2 + -2z2

Combine like terms: 2z2 + -2z2 = 0
2 + 18z + -2z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + 9z + -1z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + 9z + -1z2)' equal to zero and attempt to solve:

Simplifying
1 + 9z + -1z2 = 0

Solving
1 + 9z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-1 + -9z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -9z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -9z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -9z + z2 = 0 + 1
-9z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-9z + z2 = 1

The z term is -9z.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9z + 20.25 + z2 = 1 + 20.25

Reorder the terms:
20.25 + -9z + z2 = 1 + 20.25

Combine like terms: 1 + 20.25 = 21.25
20.25 + -9z + z2 = 21.25

Factor a perfect square on the left side:
(z + -4.5)(z + -4.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(z + -4.5) equal to 4.609772229 and -4.609772229.

Subproblem 1
z + -4.5 = 4.609772229

Simplifying
z + -4.5 = 4.609772229

Reorder the terms:
-4.5 + z = 4.609772229

Solving
-4.5 + z = 4.609772229

Solving for variable 'z'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + z = 4.609772229 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + z = 4.609772229 + 4.5
z = 4.609772229 + 4.5

Combine like terms: 4.609772229 + 4.5 = 9.109772229
z = 9.109772229

Simplifying
z = 9.109772229

Subproblem 2
z + -4.5 = -4.609772229

Simplifying
z + -4.5 = -4.609772229

Reorder the terms:
-4.5 + z = -4.609772229

Solving
-4.5 + z = -4.609772229

Solving for variable 'z'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + z = -4.609772229 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + z = -4.609772229 + 4.5
z = -4.609772229 + 4.5

Combine like terms: -4.609772229 + 4.5 = -0.109772229
z = -0.109772229

Simplifying
z = -0.109772229

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {9.109772229, -0.109772229}
Solution
z = {9.109772229, -0.109772229}