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

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

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

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

Reorder the terms for easier multiplication:
12 * 8n(-7 + 3n) = -2(4 + -3n)

Multiply 12 * 8
96n(-7 + 3n) = -2(4 + -3n)
(-7 * 96n + 3n * 96n) = -2(4 + -3n)
(-672n + 288n2) = -2(4 + -3n)
-672n + 288n2 = (4 * -2 + -3n * -2)
-672n + 288n2 = (-8 + 6n)

Solving
-672n + 288n2 = -8 + 6n

Solving for variable 'n'.

Reorder the terms:
8 + -672n + -6n + 288n2 = -8 + 6n + 8 + -6n

Combine like terms: -672n + -6n = -678n
8 + -678n + 288n2 = -8 + 6n + 8 + -6n

Reorder the terms:
8 + -678n + 288n2 = -8 + 8 + 6n + -6n

Combine like terms: -8 + 8 = 0
8 + -678n + 288n2 = 0 + 6n + -6n
8 + -678n + 288n2 = 6n + -6n

Combine like terms: 6n + -6n = 0
8 + -678n + 288n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(4 + -339n + 144n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(4 + -339n + 144n2)' equal to zero and attempt to solve:

Simplifying
4 + -339n + 144n2 = 0

Solving
4 + -339n + 144n2 = 0

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

Divide each side by '144'.
0.02777777778 + -2.354166667n + n2 = 0

Move the constant term to the right:

Add '-0.02777777778' to each side of the equation.
0.02777777778 + -2.354166667n + -0.02777777778 + n2 = 0 + -0.02777777778

Reorder the terms:
0.02777777778 + -0.02777777778 + -2.354166667n + n2 = 0 + -0.02777777778

Combine like terms: 0.02777777778 + -0.02777777778 = 0.00000000000
0.00000000000 + -2.354166667n + n2 = 0 + -0.02777777778
-2.354166667n + n2 = 0 + -0.02777777778

Combine like terms: 0 + -0.02777777778 = -0.02777777778
-2.354166667n + n2 = -0.02777777778

The n term is -2.354166667n.  Take half its coefficient (-1.177083334).
Square it (1.385525175) and add it to both sides.

Add '1.385525175' to each side of the equation.
-2.354166667n + 1.385525175 + n2 = -0.02777777778 + 1.385525175

Reorder the terms:
1.385525175 + -2.354166667n + n2 = -0.02777777778 + 1.385525175

Combine like terms: -0.02777777778 + 1.385525175 = 1.35774739722
1.385525175 + -2.354166667n + n2 = 1.35774739722

Factor a perfect square on the left side:
(n + -1.177083334)(n + -1.177083334) = 1.35774739722

Calculate the square root of the right side: 1.165224183

Break this problem into two subproblems by setting 
(n + -1.177083334) equal to 1.165224183 and -1.165224183.

Subproblem 1
n + -1.177083334 = 1.165224183

Simplifying
n + -1.177083334 = 1.165224183

Reorder the terms:
-1.177083334 + n = 1.165224183

Solving
-1.177083334 + n = 1.165224183

Solving for variable 'n'.

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

Add '1.177083334' to each side of the equation.
-1.177083334 + 1.177083334 + n = 1.165224183 + 1.177083334

Combine like terms: -1.177083334 + 1.177083334 = 0.000000000
0.000000000 + n = 1.165224183 + 1.177083334
n = 1.165224183 + 1.177083334

Combine like terms: 1.165224183 + 1.177083334 = 2.342307517
n = 2.342307517

Simplifying
n = 2.342307517

Subproblem 2
n + -1.177083334 = -1.165224183

Simplifying
n + -1.177083334 = -1.165224183

Reorder the terms:
-1.177083334 + n = -1.165224183

Solving
-1.177083334 + n = -1.165224183

Solving for variable 'n'.

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

Add '1.177083334' to each side of the equation.
-1.177083334 + 1.177083334 + n = -1.165224183 + 1.177083334

Combine like terms: -1.177083334 + 1.177083334 = 0.000000000
0.000000000 + n = -1.165224183 + 1.177083334
n = -1.165224183 + 1.177083334

Combine like terms: -1.165224183 + 1.177083334 = 0.011859151
n = 0.011859151

Simplifying
n = 0.011859151

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.342307517, 0.011859151}
Solution
n = {2.342307517, 0.011859151}