(6-2n)8n=34

Solution for (6-2n)8n=34 equation:

Simplifying
(6 + -2n) * 8n = 34

Reorder the terms for easier multiplication:
8n(6 + -2n) = 34
(6 * 8n + -2n * 8n) = 34
(48n + -16n2) = 34

Solving
48n + -16n2 = 34

Solving for variable 'n'.

Reorder the terms:
-34 + 48n + -16n2 = 34 + -34

Combine like terms: 34 + -34 = 0
-34 + 48n + -16n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-17 + 24n + -8n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-17 + 24n + -8n2)' equal to zero and attempt to solve:

Simplifying
-17 + 24n + -8n2 = 0

Solving
-17 + 24n + -8n2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
2.125 + -3n + n2 = 0

Move the constant term to the right:

Add '-2.125' to each side of the equation.
2.125 + -3n + -2.125 + n2 = 0 + -2.125

Reorder the terms:
2.125 + -2.125 + -3n + n2 = 0 + -2.125

Combine like terms: 2.125 + -2.125 = 0.000
0.000 + -3n + n2 = 0 + -2.125
-3n + n2 = 0 + -2.125

Combine like terms: 0 + -2.125 = -2.125
-3n + n2 = -2.125

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 = -2.125 + 2.25

Reorder the terms:
2.25 + -3n + n2 = -2.125 + 2.25

Combine like terms: -2.125 + 2.25 = 0.125
2.25 + -3n + n2 = 0.125

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 0.125

Calculate the square root of the right side: 0.353553391

Break this problem into two subproblems by setting 
(n + -1.5) equal to 0.353553391 and -0.353553391.

Subproblem 1
n + -1.5 = 0.353553391

Simplifying
n + -1.5 = 0.353553391

Reorder the terms:
-1.5 + n = 0.353553391

Solving
-1.5 + n = 0.353553391

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 = 0.353553391 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 0.353553391 + 1.5
n = 0.353553391 + 1.5

Combine like terms: 0.353553391 + 1.5 = 1.853553391
n = 1.853553391

Simplifying
n = 1.853553391

Subproblem 2
n + -1.5 = -0.353553391

Simplifying
n + -1.5 = -0.353553391

Reorder the terms:
-1.5 + n = -0.353553391

Solving
-1.5 + n = -0.353553391

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 = -0.353553391 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -0.353553391 + 1.5
n = -0.353553391 + 1.5

Combine like terms: -0.353553391 + 1.5 = 1.146446609
n = 1.146446609

Simplifying
n = 1.146446609

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.853553391, 1.146446609}
Solution
n = {1.853553391, 1.146446609}