52+4(z-2)=2(z+3)(z-2)

Solution for 52+4(z-2)=2(z+3)(z-2) equation:

Simplifying
52 + 4(z + -2) = 2(z + 3)(z + -2)

Reorder the terms:
52 + 4(-2 + z) = 2(z + 3)(z + -2)
52 + (-2 * 4 + z * 4) = 2(z + 3)(z + -2)
52 + (-8 + 4z) = 2(z + 3)(z + -2)

Combine like terms: 52 + -8 = 44
44 + 4z = 2(z + 3)(z + -2)

Reorder the terms:
44 + 4z = 2(3 + z)(z + -2)

Reorder the terms:
44 + 4z = 2(3 + z)(-2 + z)

Multiply (3 + z) * (-2 + z)
44 + 4z = 2(3(-2 + z) + z(-2 + z))
44 + 4z = 2((-2 * 3 + z * 3) + z(-2 + z))
44 + 4z = 2((-6 + 3z) + z(-2 + z))
44 + 4z = 2(-6 + 3z + (-2 * z + z * z))
44 + 4z = 2(-6 + 3z + (-2z + z2))

Combine like terms: 3z + -2z = 1z
44 + 4z = 2(-6 + 1z + z2)
44 + 4z = (-6 * 2 + 1z * 2 + z2 * 2)
44 + 4z = (-12 + 2z + 2z2)

Solving
44 + 4z = -12 + 2z + 2z2

Solving for variable 'z'.

Reorder the terms:
44 + 12 + 4z + -2z + -2z2 = -12 + 2z + 2z2 + 12 + -2z + -2z2

Combine like terms: 44 + 12 = 56
56 + 4z + -2z + -2z2 = -12 + 2z + 2z2 + 12 + -2z + -2z2

Combine like terms: 4z + -2z = 2z
56 + 2z + -2z2 = -12 + 2z + 2z2 + 12 + -2z + -2z2

Reorder the terms:
56 + 2z + -2z2 = -12 + 12 + 2z + -2z + 2z2 + -2z2

Combine like terms: -12 + 12 = 0
56 + 2z + -2z2 = 0 + 2z + -2z + 2z2 + -2z2
56 + 2z + -2z2 = 2z + -2z + 2z2 + -2z2

Combine like terms: 2z + -2z = 0
56 + 2z + -2z2 = 0 + 2z2 + -2z2
56 + 2z + -2z2 = 2z2 + -2z2

Combine like terms: 2z2 + -2z2 = 0
56 + 2z + -2z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(28 + z + -1z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(28 + z + -1z2)' equal to zero and attempt to solve:

Simplifying
28 + z + -1z2 = 0

Solving
28 + z + -1z2 = 0

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

Divide each side by '-1'.
-28 + -1z + z2 = 0

Move the constant term to the right:

Add '28' to each side of the equation.
-28 + -1z + 28 + z2 = 0 + 28

Reorder the terms:
-28 + 28 + -1z + z2 = 0 + 28

Combine like terms: -28 + 28 = 0
0 + -1z + z2 = 0 + 28
-1z + z2 = 0 + 28

Combine like terms: 0 + 28 = 28
-1z + z2 = 28

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.
-1z + 0.25 + z2 = 28 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 28 + 0.25

Combine like terms: 28 + 0.25 = 28.25
0.25 + -1z + z2 = 28.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 28.25

Calculate the square root of the right side: 5.315072906

Break this problem into two subproblems by setting 
(z + 0.5) equal to 5.315072906 and -5.315072906.

Subproblem 1
z + 0.5 = 5.315072906

Simplifying
z + 0.5 = 5.315072906

Reorder the terms:
0.5 + z = 5.315072906

Solving
0.5 + z = 5.315072906

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 = 5.315072906 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = 5.315072906 + -0.5
z = 5.315072906 + -0.5

Combine like terms: 5.315072906 + -0.5 = 4.815072906
z = 4.815072906

Simplifying
z = 4.815072906

Subproblem 2
z + 0.5 = -5.315072906

Simplifying
z + 0.5 = -5.315072906

Reorder the terms:
0.5 + z = -5.315072906

Solving
0.5 + z = -5.315072906

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 = -5.315072906 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + z = -5.315072906 + -0.5
z = -5.315072906 + -0.5

Combine like terms: -5.315072906 + -0.5 = -5.815072906
z = -5.815072906

Simplifying
z = -5.815072906

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.815072906, -5.815072906}
Solution
z = {4.815072906, -5.815072906}