12b^2-12b-1=7

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Solution for 12b^2-12b-1=7 equation: 


Simplifying
12b2 + -12b + -1 = 7

Reorder the terms:
-1 + -12b + 12b2 = 7

Solving
-1 + -12b + 12b2 = 7

Solving for variable 'b'.

Reorder the terms:
-1 + -7 + -12b + 12b2 = 7 + -7

Combine like terms: -1 + -7 = -8
-8 + -12b + 12b2 = 7 + -7

Combine like terms: 7 + -7 = 0
-8 + -12b + 12b2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-2 + -3b + 3b2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-2 + -3b + 3b2)' equal to zero and attempt to solve:

Simplifying
-2 + -3b + 3b2 = 0

Solving
-2 + -3b + 3b2 = 0

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

Divide each side by '3'.
-0.6666666667 + -1b + b2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + -1b + 0.6666666667 + b2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + -1b + b2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + -1b + b2 = 0 + 0.6666666667
-1b + b2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
-1b + b2 = 0.6666666667

The b term is -1b.  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.
-1b + 0.25 + b2 = 0.6666666667 + 0.25

Reorder the terms:
0.25 + -1b + b2 = 0.6666666667 + 0.25

Combine like terms: 0.6666666667 + 0.25 = 0.9166666667
0.25 + -1b + b2 = 0.9166666667

Factor a perfect square on the left side:
(b + -0.5)(b + -0.5) = 0.9166666667

Calculate the square root of the right side: 0.957427108

Break this problem into two subproblems by setting 
(b + -0.5) equal to 0.957427108 and -0.957427108.


Subproblem 1
b + -0.5 = 0.957427108

Simplifying
b + -0.5 = 0.957427108

Reorder the terms:
-0.5 + b = 0.957427108

Solving
-0.5 + b = 0.957427108

Solving for variable 'b'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + b = 0.957427108 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + b = 0.957427108 + 0.5
b = 0.957427108 + 0.5

Combine like terms: 0.957427108 + 0.5 = 1.457427108
b = 1.457427108

Simplifying
b = 1.457427108


Subproblem 2
b + -0.5 = -0.957427108

Simplifying
b + -0.5 = -0.957427108

Reorder the terms:
-0.5 + b = -0.957427108

Solving
-0.5 + b = -0.957427108

Solving for variable 'b'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + b = -0.957427108 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + b = -0.957427108 + 0.5
b = -0.957427108 + 0.5

Combine like terms: -0.957427108 + 0.5 = -0.457427108
b = -0.457427108

Simplifying
b = -0.457427108


Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {1.457427108, -0.457427108}
Solution
b = {1.457427108, -0.457427108}

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