12n(3n-7)+8n=-2(4-3n)

Solution for 12n(3n-7)+8n=-2(4-3n) equation:

Simplifying
12n(3n + -7) + 8n = -2(4 + -3n)

Reorder the terms:
12n(-7 + 3n) + 8n = -2(4 + -3n)
(-7 * 12n + 3n * 12n) + 8n = -2(4 + -3n)
(-84n + 36n2) + 8n = -2(4 + -3n)

Reorder the terms:
-84n + 8n + 36n2 = -2(4 + -3n)

Combine like terms: -84n + 8n = -76n
-76n + 36n2 = -2(4 + -3n)
-76n + 36n2 = (4 * -2 + -3n * -2)
-76n + 36n2 = (-8 + 6n)

Solving
-76n + 36n2 = -8 + 6n

Solving for variable 'n'.

Reorder the terms:
8 + -76n + -6n + 36n2 = -8 + 6n + 8 + -6n

Combine like terms: -76n + -6n = -82n
8 + -82n + 36n2 = -8 + 6n + 8 + -6n

Reorder the terms:
8 + -82n + 36n2 = -8 + 8 + 6n + -6n

Combine like terms: -8 + 8 = 0
8 + -82n + 36n2 = 0 + 6n + -6n
8 + -82n + 36n2 = 6n + -6n

Combine like terms: 6n + -6n = 0
8 + -82n + 36n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(4 + -41n + 18n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(4 + -41n + 18n2)' equal to zero and attempt to solve:

Simplifying
4 + -41n + 18n2 = 0

Solving
4 + -41n + 18n2 = 0

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

Divide each side by '18'.
0.2222222222 + -2.277777778n + n2 = 0

Move the constant term to the right:

Add '-0.2222222222' to each side of the equation.
0.2222222222 + -2.277777778n + -0.2222222222 + n2 = 0 + -0.2222222222

Reorder the terms:
0.2222222222 + -0.2222222222 + -2.277777778n + n2 = 0 + -0.2222222222

Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000
0.0000000000 + -2.277777778n + n2 = 0 + -0.2222222222
-2.277777778n + n2 = 0 + -0.2222222222

Combine like terms: 0 + -0.2222222222 = -0.2222222222
-2.277777778n + n2 = -0.2222222222

The n term is -2.277777778n.  Take half its coefficient (-1.138888889).
Square it (1.297067901) and add it to both sides.

Add '1.297067901' to each side of the equation.
-2.277777778n + 1.297067901 + n2 = -0.2222222222 + 1.297067901

Reorder the terms:
1.297067901 + -2.277777778n + n2 = -0.2222222222 + 1.297067901

Combine like terms: -0.2222222222 + 1.297067901 = 1.0748456788
1.297067901 + -2.277777778n + n2 = 1.0748456788

Factor a perfect square on the left side:
(n + -1.138888889)(n + -1.138888889) = 1.0748456788

Calculate the square root of the right side: 1.036747645

Break this problem into two subproblems by setting 
(n + -1.138888889) equal to 1.036747645 and -1.036747645.

Subproblem 1
n + -1.138888889 = 1.036747645

Simplifying
n + -1.138888889 = 1.036747645

Reorder the terms:
-1.138888889 + n = 1.036747645

Solving
-1.138888889 + n = 1.036747645

Solving for variable 'n'.

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

Add '1.138888889' to each side of the equation.
-1.138888889 + 1.138888889 + n = 1.036747645 + 1.138888889

Combine like terms: -1.138888889 + 1.138888889 = 0.000000000
0.000000000 + n = 1.036747645 + 1.138888889
n = 1.036747645 + 1.138888889

Combine like terms: 1.036747645 + 1.138888889 = 2.175636534
n = 2.175636534

Simplifying
n = 2.175636534

Subproblem 2
n + -1.138888889 = -1.036747645

Simplifying
n + -1.138888889 = -1.036747645

Reorder the terms:
-1.138888889 + n = -1.036747645

Solving
-1.138888889 + n = -1.036747645

Solving for variable 'n'.

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

Add '1.138888889' to each side of the equation.
-1.138888889 + 1.138888889 + n = -1.036747645 + 1.138888889

Combine like terms: -1.138888889 + 1.138888889 = 0.000000000
0.000000000 + n = -1.036747645 + 1.138888889
n = -1.036747645 + 1.138888889

Combine like terms: -1.036747645 + 1.138888889 = 0.102141244
n = 0.102141244

Simplifying
n = 0.102141244

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.175636534, 0.102141244}
Solution
n = {2.175636534, 0.102141244}