5z(z+2)=5(z+1)

Solution for 5z(z+2)=5(z+1) equation:

Simplifying
5z(z + 2) = 5(z + 1)

Reorder the terms:
5z(2 + z) = 5(z + 1)
(2 * 5z + z * 5z) = 5(z + 1)
(10z + 5z2) = 5(z + 1)

Reorder the terms:
10z + 5z2 = 5(1 + z)
10z + 5z2 = (1 * 5 + z * 5)
10z + 5z2 = (5 + 5z)

Solving
10z + 5z2 = 5 + 5z

Solving for variable 'z'.

Reorder the terms:
-5 + 10z + -5z + 5z2 = 5 + 5z + -5 + -5z

Combine like terms: 10z + -5z = 5z
-5 + 5z + 5z2 = 5 + 5z + -5 + -5z

Reorder the terms:
-5 + 5z + 5z2 = 5 + -5 + 5z + -5z

Combine like terms: 5 + -5 = 0
-5 + 5z + 5z2 = 0 + 5z + -5z
-5 + 5z + 5z2 = 5z + -5z

Combine like terms: 5z + -5z = 0
-5 + 5z + 5z2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-1 + z + z2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-1 + z + z2)' equal to zero and attempt to solve:

Simplifying
-1 + z + z2 = 0

Solving
-1 + z + z2 = 0

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
z + 0.25 + z2 = 1 + 0.25

Reorder the terms:
0.25 + z + z2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + z + z2 = 1.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
z + 0.5 = 1.118033989

Simplifying
z + 0.5 = 1.118033989

Reorder the terms:
0.5 + z = 1.118033989

Solving
0.5 + z = 1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 1.118033989 + -0.5
z = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
z = 0.618033989

Simplifying
z = 0.618033989

Subproblem 2
z + 0.5 = -1.118033989

Simplifying
z + 0.5 = -1.118033989

Reorder the terms:
0.5 + z = -1.118033989

Solving
0.5 + z = -1.118033989

Solving for variable 'z'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + z = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -1.118033989 + -0.5
z = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
z = -1.618033989

Simplifying
z = -1.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.618033989, -1.618033989}
Solution
z = {0.618033989, -1.618033989}