8n-2(n^2+n)+12=

Solution for 8n-2(n^2+n)+12= equation:

Simplifying
8n + -2(n2 + n) + 12 = 0

Reorder the terms:
8n + -2(n + n2) + 12 = 0
8n + (n * -2 + n2 * -2) + 12 = 0
8n + (-2n + -2n2) + 12 = 0

Reorder the terms:
12 + 8n + -2n + -2n2 = 0

Combine like terms: 8n + -2n = 6n
12 + 6n + -2n2 = 0

Solving
12 + 6n + -2n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(6 + 3n + -1n2) = 0

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

Simplifying
6 + 3n + -1n2 = 0

Solving
6 + 3n + -1n2 = 0

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

Divide each side by '-1'.
-6 + -3n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -3n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + -3n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -3n + n2 = 0 + 6
-3n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
-3n + n2 = 6

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 6 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + -3n + n2 = 8.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(n + -1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1
n + -1.5 = 2.872281323

Simplifying
n + -1.5 = 2.872281323

Reorder the terms:
-1.5 + n = 2.872281323

Solving
-1.5 + n = 2.872281323

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 2.872281323 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 2.872281323 + 1.5
n = 2.872281323 + 1.5

Combine like terms: 2.872281323 + 1.5 = 4.372281323
n = 4.372281323

Simplifying
n = 4.372281323

Subproblem 2
n + -1.5 = -2.872281323

Simplifying
n + -1.5 = -2.872281323

Reorder the terms:
-1.5 + n = -2.872281323

Solving
-1.5 + n = -2.872281323

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -2.872281323 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -2.872281323 + 1.5
n = -2.872281323 + 1.5

Combine like terms: -2.872281323 + 1.5 = -1.372281323
n = -1.372281323

Simplifying
n = -1.372281323

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.372281323, -1.372281323}
Solution
n = {4.372281323, -1.372281323}