6n^2+2n-192=0

Solution for 6n^2+2n-192=0 equation:

Simplifying
6n2 + 2n + -192 = 0

Reorder the terms:
-192 + 2n + 6n2 = 0

Solving
-192 + 2n + 6n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-96 + n + 3n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-96 + n + 3n2)' equal to zero and attempt to solve:

Simplifying
-96 + n + 3n2 = 0

Solving
-96 + n + 3n2 = 0

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

Divide each side by '3'.
-32 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '32' to each side of the equation.
-32 + 0.3333333333n + 32 + n2 = 0 + 32

Reorder the terms:
-32 + 32 + 0.3333333333n + n2 = 0 + 32

Combine like terms: -32 + 32 = 0
0 + 0.3333333333n + n2 = 0 + 32
0.3333333333n + n2 = 0 + 32

Combine like terms: 0 + 32 = 32
0.3333333333n + n2 = 32

The n term is n.  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.
0.3333333333n + 0.25 + n2 = 32 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 32 + 0.25

Combine like terms: 32 + 0.25 = 32.25
0.25 + 0.3333333333n + n2 = 32.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 32.25

Calculate the square root of the right side: 5.678908346

Break this problem into two subproblems by setting 
(n + 0.5) equal to 5.678908346 and -5.678908346.

Subproblem 1
n + 0.5 = 5.678908346

Simplifying
n + 0.5 = 5.678908346

Reorder the terms:
0.5 + n = 5.678908346

Solving
0.5 + n = 5.678908346

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 5.678908346 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 5.678908346 + -0.5
n = 5.678908346 + -0.5

Combine like terms: 5.678908346 + -0.5 = 5.178908346
n = 5.178908346

Simplifying
n = 5.178908346

Subproblem 2
n + 0.5 = -5.678908346

Simplifying
n + 0.5 = -5.678908346

Reorder the terms:
0.5 + n = -5.678908346

Solving
0.5 + n = -5.678908346

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -5.678908346 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -5.678908346 + -0.5
n = -5.678908346 + -0.5

Combine like terms: -5.678908346 + -0.5 = -6.178908346
n = -6.178908346

Simplifying
n = -6.178908346

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.178908346, -6.178908346}
Solution
n = {5.178908346, -6.178908346}