8v^2-8v-12=0

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Solution for 8v^2-8v-12=0 equation: 


Simplifying
8v2 + -8v + -12 = 0

Reorder the terms:
-12 + -8v + 8v2 = 0

Solving
-12 + -8v + 8v2 = 0

Solving for variable 'v'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-3 + -2v + 2v2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-3 + -2v + 2v2)' equal to zero and attempt to solve:

Simplifying
-3 + -2v + 2v2 = 0

Solving
-3 + -2v + 2v2 = 0

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

Divide each side by '2'.
-1.5 + -1v + v2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -1v + 1.5 + v2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -1v + v2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -1v + v2 = 0 + 1.5
-1v + v2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-1v + v2 = 1.5

The v term is -1v.  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.
-1v + 0.25 + v2 = 1.5 + 0.25

Reorder the terms:
0.25 + -1v + v2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + -1v + v2 = 1.75

Factor a perfect square on the left side:
(v + -0.5)(v + -0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(v + -0.5) equal to 1.322875656 and -1.322875656.


Subproblem 1
v + -0.5 = 1.322875656

Simplifying
v + -0.5 = 1.322875656

Reorder the terms:
-0.5 + v = 1.322875656

Solving
-0.5 + v = 1.322875656

Solving for variable 'v'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + v = 1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + v = 1.322875656 + 0.5
v = 1.322875656 + 0.5

Combine like terms: 1.322875656 + 0.5 = 1.822875656
v = 1.822875656

Simplifying
v = 1.822875656


Subproblem 2
v + -0.5 = -1.322875656

Simplifying
v + -0.5 = -1.322875656

Reorder the terms:
-0.5 + v = -1.322875656

Solving
-0.5 + v = -1.322875656

Solving for variable 'v'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + v = -1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + v = -1.322875656 + 0.5
v = -1.322875656 + 0.5

Combine like terms: -1.322875656 + 0.5 = -0.822875656
v = -0.822875656

Simplifying
v = -0.822875656


Solution
The solution to the problem is based on the solutions
from the subproblems.
v = {1.822875656, -0.822875656}
Solution
v = {1.822875656, -0.822875656}

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