2n(2n)-6n=8

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

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

Remove parenthesis around (2n)
2n * 2n + -6n = 8

Reorder the terms for easier multiplication:
2 * 2n * n + -6n = 8

Multiply 2 * 2
4n * n + -6n = 8

Multiply n * n
4n2 + -6n = 8

Reorder the terms:
-6n + 4n2 = 8

Solving
-6n + 4n2 = 8

Solving for variable 'n'.

Reorder the terms:
-8 + -6n + 4n2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + -6n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + -3n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-4 + -3n + 2n2)' equal to zero and attempt to solve:

Simplifying
-4 + -3n + 2n2 = 0

Solving
-4 + -3n + 2n2 = 0

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

Divide each side by '2'.
-2 + -1.5n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.5n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.5n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.5n + n2 = 0 + 2
-1.5n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.5n + n2 = 2

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 2 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 2 + 0.5625

Combine like terms: 2 + 0.5625 = 2.5625
0.5625 + -1.5n + n2 = 2.5625

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 2.5625

Calculate the square root of the right side: 1.600781059

Break this problem into two subproblems by setting 
(n + -0.75) equal to 1.600781059 and -1.600781059.

Subproblem 1
n + -0.75 = 1.600781059

Simplifying
n + -0.75 = 1.600781059

Reorder the terms:
-0.75 + n = 1.600781059

Solving
-0.75 + n = 1.600781059

Solving for variable 'n'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = 1.600781059 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = 1.600781059 + 0.75
n = 1.600781059 + 0.75

Combine like terms: 1.600781059 + 0.75 = 2.350781059
n = 2.350781059

Simplifying
n = 2.350781059

Subproblem 2
n + -0.75 = -1.600781059

Simplifying
n + -0.75 = -1.600781059

Reorder the terms:
-0.75 + n = -1.600781059

Solving
-0.75 + n = -1.600781059

Solving for variable 'n'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + n = -1.600781059 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + n = -1.600781059 + 0.75
n = -1.600781059 + 0.75

Combine like terms: -1.600781059 + 0.75 = -0.850781059
n = -0.850781059

Simplifying
n = -0.850781059

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.350781059, -0.850781059}
Solution
n = {2.350781059, -0.850781059}