12n^2=-2n+6

Solution for 12n^2=-2n+6 equation:

Simplifying
12n2 = -2n + 6

Reorder the terms:
12n2 = 6 + -2n

Solving
12n2 = 6 + -2n

Solving for variable 'n'.

Reorder the terms:
-6 + 2n + 12n2 = 6 + -2n + -6 + 2n

Reorder the terms:
-6 + 2n + 12n2 = 6 + -6 + -2n + 2n

Combine like terms: 6 + -6 = 0
-6 + 2n + 12n2 = 0 + -2n + 2n
-6 + 2n + 12n2 = -2n + 2n

Combine like terms: -2n + 2n = 0
-6 + 2n + 12n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + n + 6n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + n + 6n2)' equal to zero and attempt to solve:

Simplifying
-3 + n + 6n2 = 0

Solving
-3 + n + 6n2 = 0

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

Divide each side by '6'.
-0.5 + 0.1666666667n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.1666666667n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.1666666667n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.1666666667n + n2 = 0 + 0.5
0.1666666667n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.1666666667n + n2 = 0.5

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.
0.1666666667n + 0.25 + n2 = 0.5 + 0.25

Reorder the terms:
0.25 + 0.1666666667n + n2 = 0.5 + 0.25

Combine like terms: 0.5 + 0.25 = 0.75
0.25 + 0.1666666667n + n2 = 0.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
n + 0.5 = 0.866025404

Simplifying
n + 0.5 = 0.866025404

Reorder the terms:
0.5 + n = 0.866025404

Solving
0.5 + n = 0.866025404

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 = 0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 0.866025404 + -0.5
n = 0.866025404 + -0.5

Combine like terms: 0.866025404 + -0.5 = 0.366025404
n = 0.366025404

Simplifying
n = 0.366025404

Subproblem 2
n + 0.5 = -0.866025404

Simplifying
n + 0.5 = -0.866025404

Reorder the terms:
0.5 + n = -0.866025404

Solving
0.5 + n = -0.866025404

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 = -0.866025404 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -0.866025404 + -0.5
n = -0.866025404 + -0.5

Combine like terms: -0.866025404 + -0.5 = -1.366025404
n = -1.366025404

Simplifying
n = -1.366025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.366025404, -1.366025404}
Solution
n = {0.366025404, -1.366025404}