11(z+1)4(z-4)=4(z-1)+2(z-2)

Solution for 11(z+1)4(z-4)=4(z-1)+2(z-2) equation:

Simplifying
11(z + 1) * 4(z + -4) = 4(z + -1) + 2(z + -2)

Reorder the terms:
11(1 + z) * 4(z + -4) = 4(z + -1) + 2(z + -2)

Reorder the terms:
11(1 + z) * 4(-4 + z) = 4(z + -1) + 2(z + -2)

Reorder the terms for easier multiplication:
11 * 4(1 + z)(-4 + z) = 4(z + -1) + 2(z + -2)

Multiply 11 * 4
44(1 + z)(-4 + z) = 4(z + -1) + 2(z + -2)

Multiply (1 + z) * (-4 + z)
44(1(-4 + z) + z(-4 + z)) = 4(z + -1) + 2(z + -2)
44((-4 * 1 + z * 1) + z(-4 + z)) = 4(z + -1) + 2(z + -2)
44((-4 + 1z) + z(-4 + z)) = 4(z + -1) + 2(z + -2)
44(-4 + 1z + (-4 * z + z * z)) = 4(z + -1) + 2(z + -2)
44(-4 + 1z + (-4z + z2)) = 4(z + -1) + 2(z + -2)

Combine like terms: 1z + -4z = -3z
44(-4 + -3z + z2) = 4(z + -1) + 2(z + -2)
(-4 * 44 + -3z * 44 + z2 * 44) = 4(z + -1) + 2(z + -2)
(-176 + -132z + 44z2) = 4(z + -1) + 2(z + -2)

Reorder the terms:
-176 + -132z + 44z2 = 4(-1 + z) + 2(z + -2)
-176 + -132z + 44z2 = (-1 * 4 + z * 4) + 2(z + -2)
-176 + -132z + 44z2 = (-4 + 4z) + 2(z + -2)

Reorder the terms:
-176 + -132z + 44z2 = -4 + 4z + 2(-2 + z)
-176 + -132z + 44z2 = -4 + 4z + (-2 * 2 + z * 2)
-176 + -132z + 44z2 = -4 + 4z + (-4 + 2z)

Reorder the terms:
-176 + -132z + 44z2 = -4 + -4 + 4z + 2z

Combine like terms: -4 + -4 = -8
-176 + -132z + 44z2 = -8 + 4z + 2z

Combine like terms: 4z + 2z = 6z
-176 + -132z + 44z2 = -8 + 6z

Solving
-176 + -132z + 44z2 = -8 + 6z

Solving for variable 'z'.

Reorder the terms:
-176 + 8 + -132z + -6z + 44z2 = -8 + 6z + 8 + -6z

Combine like terms: -176 + 8 = -168
-168 + -132z + -6z + 44z2 = -8 + 6z + 8 + -6z

Combine like terms: -132z + -6z = -138z
-168 + -138z + 44z2 = -8 + 6z + 8 + -6z

Reorder the terms:
-168 + -138z + 44z2 = -8 + 8 + 6z + -6z

Combine like terms: -8 + 8 = 0
-168 + -138z + 44z2 = 0 + 6z + -6z
-168 + -138z + 44z2 = 6z + -6z

Combine like terms: 6z + -6z = 0
-168 + -138z + 44z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-84 + -69z + 22z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-84 + -69z + 22z2)' equal to zero and attempt to solve:

Simplifying
-84 + -69z + 22z2 = 0

Solving
-84 + -69z + 22z2 = 0

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

Divide each side by '22'.
-3.818181818 + -3.136363636z + z2 = 0

Move the constant term to the right:

Add '3.818181818' to each side of the equation.
-3.818181818 + -3.136363636z + 3.818181818 + z2 = 0 + 3.818181818

Reorder the terms:
-3.818181818 + 3.818181818 + -3.136363636z + z2 = 0 + 3.818181818

Combine like terms: -3.818181818 + 3.818181818 = 0.000000000
0.000000000 + -3.136363636z + z2 = 0 + 3.818181818
-3.136363636z + z2 = 0 + 3.818181818

Combine like terms: 0 + 3.818181818 = 3.818181818
-3.136363636z + z2 = 3.818181818

The z term is -3.136363636z.  Take half its coefficient (-1.568181818).
Square it (2.459194214) and add it to both sides.

Add '2.459194214' to each side of the equation.
-3.136363636z + 2.459194214 + z2 = 3.818181818 + 2.459194214

Reorder the terms:
2.459194214 + -3.136363636z + z2 = 3.818181818 + 2.459194214

Combine like terms: 3.818181818 + 2.459194214 = 6.277376032
2.459194214 + -3.136363636z + z2 = 6.277376032

Factor a perfect square on the left side:
(z + -1.568181818)(z + -1.568181818) = 6.277376032

Calculate the square root of the right side: 2.505469224

Break this problem into two subproblems by setting 
(z + -1.568181818) equal to 2.505469224 and -2.505469224.

Subproblem 1
z + -1.568181818 = 2.505469224

Simplifying
z + -1.568181818 = 2.505469224

Reorder the terms:
-1.568181818 + z = 2.505469224

Solving
-1.568181818 + z = 2.505469224

Solving for variable 'z'.

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

Add '1.568181818' to each side of the equation.
-1.568181818 + 1.568181818 + z = 2.505469224 + 1.568181818

Combine like terms: -1.568181818 + 1.568181818 = 0.000000000
0.000000000 + z = 2.505469224 + 1.568181818
z = 2.505469224 + 1.568181818

Combine like terms: 2.505469224 + 1.568181818 = 4.073651042
z = 4.073651042

Simplifying
z = 4.073651042

Subproblem 2
z + -1.568181818 = -2.505469224

Simplifying
z + -1.568181818 = -2.505469224

Reorder the terms:
-1.568181818 + z = -2.505469224

Solving
-1.568181818 + z = -2.505469224

Solving for variable 'z'.

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

Add '1.568181818' to each side of the equation.
-1.568181818 + 1.568181818 + z = -2.505469224 + 1.568181818

Combine like terms: -1.568181818 + 1.568181818 = 0.000000000
0.000000000 + z = -2.505469224 + 1.568181818
z = -2.505469224 + 1.568181818

Combine like terms: -2.505469224 + 1.568181818 = -0.937287406
z = -0.937287406

Simplifying
z = -0.937287406

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.073651042, -0.937287406}
Solution
z = {4.073651042, -0.937287406}