12Z+12-Z^2-1=20Z-20

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Solution for 12Z+12-Z^2-1=20Z-20 equation: 


Simplifying
12Z + 12 + -1Z2 + -1 = 20Z + -20

Reorder the terms:
12 + -1 + 12Z + -1Z2 = 20Z + -20

Combine like terms: 12 + -1 = 11
11 + 12Z + -1Z2 = 20Z + -20

Reorder the terms:
11 + 12Z + -1Z2 = -20 + 20Z

Solving
11 + 12Z + -1Z2 = -20 + 20Z

Solving for variable 'Z'.

Reorder the terms:
11 + 20 + 12Z + -20Z + -1Z2 = -20 + 20Z + 20 + -20Z

Combine like terms: 11 + 20 = 31
31 + 12Z + -20Z + -1Z2 = -20 + 20Z + 20 + -20Z

Combine like terms: 12Z + -20Z = -8Z
31 + -8Z + -1Z2 = -20 + 20Z + 20 + -20Z

Reorder the terms:
31 + -8Z + -1Z2 = -20 + 20 + 20Z + -20Z

Combine like terms: -20 + 20 = 0
31 + -8Z + -1Z2 = 0 + 20Z + -20Z
31 + -8Z + -1Z2 = 20Z + -20Z

Combine like terms: 20Z + -20Z = 0
31 + -8Z + -1Z2 = 0

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

Divide each side by '-1'.
-31 + 8Z + Z2 = 0

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 8Z + 31 + Z2 = 0 + 31

Reorder the terms:
-31 + 31 + 8Z + Z2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 8Z + Z2 = 0 + 31
8Z + Z2 = 0 + 31

Combine like terms: 0 + 31 = 31
8Z + Z2 = 31

The Z term is 8Z.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8Z + 16 + Z2 = 31 + 16

Reorder the terms:
16 + 8Z + Z2 = 31 + 16

Combine like terms: 31 + 16 = 47
16 + 8Z + Z2 = 47

Factor a perfect square on the left side:
(Z + 4)(Z + 4) = 47

Calculate the square root of the right side: 6.8556546

Break this problem into two subproblems by setting 
(Z + 4) equal to 6.8556546 and -6.8556546.


Subproblem 1
Z + 4 = 6.8556546

Simplifying
Z + 4 = 6.8556546

Reorder the terms:
4 + Z = 6.8556546

Solving
4 + Z = 6.8556546

Solving for variable 'Z'.

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

Add '-4' to each side of the equation.
4 + -4 + Z = 6.8556546 + -4

Combine like terms: 4 + -4 = 0
0 + Z = 6.8556546 + -4
Z = 6.8556546 + -4

Combine like terms: 6.8556546 + -4 = 2.8556546
Z = 2.8556546

Simplifying
Z = 2.8556546


Subproblem 2
Z + 4 = -6.8556546

Simplifying
Z + 4 = -6.8556546

Reorder the terms:
4 + Z = -6.8556546

Solving
4 + Z = -6.8556546

Solving for variable 'Z'.

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

Add '-4' to each side of the equation.
4 + -4 + Z = -6.8556546 + -4

Combine like terms: 4 + -4 = 0
0 + Z = -6.8556546 + -4
Z = -6.8556546 + -4

Combine like terms: -6.8556546 + -4 = -10.8556546
Z = -10.8556546

Simplifying
Z = -10.8556546


Solution
The solution to the problem is based on the solutions
from the subproblems.
Z = {2.8556546, -10.8556546}

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