n^2-4n^2-9n+6-8=0

Solution for n^2-4n^2-9n+6-8=0 equation:

Simplifying
n2 + -4n2 + -9n + 6 + -8 = 0

Reorder the terms:
6 + -8 + -9n + n2 + -4n2 = 0

Combine like terms: 6 + -8 = -2
-2 + -9n + n2 + -4n2 = 0

Combine like terms: n2 + -4n2 = -3n2
-2 + -9n + -3n2 = 0

Solving
-2 + -9n + -3n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(2 + 9n + 3n2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(2 + 9n + 3n2)' equal to zero and attempt to solve:

Simplifying
2 + 9n + 3n2 = 0

Solving
2 + 9n + 3n2 = 0

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

Divide each side by '3'.
0.6666666667 + 3n + n2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 3n + -0.6666666667 + n2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 3n + n2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 3n + n2 = 0 + -0.6666666667
3n + n2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
3n + n2 = -0.6666666667

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 = -0.6666666667 + 2.25

Reorder the terms:
2.25 + 3n + n2 = -0.6666666667 + 2.25

Combine like terms: -0.6666666667 + 2.25 = 1.5833333333
2.25 + 3n + n2 = 1.5833333333

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 1.5833333333

Calculate the square root of the right side: 1.258305739

Break this problem into two subproblems by setting 
(n + 1.5) equal to 1.258305739 and -1.258305739.

Subproblem 1
n + 1.5 = 1.258305739

Simplifying
n + 1.5 = 1.258305739

Reorder the terms:
1.5 + n = 1.258305739

Solving
1.5 + n = 1.258305739

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 = 1.258305739 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = 1.258305739 + -1.5
n = 1.258305739 + -1.5

Combine like terms: 1.258305739 + -1.5 = -0.241694261
n = -0.241694261

Simplifying
n = -0.241694261

Subproblem 2
n + 1.5 = -1.258305739

Simplifying
n + 1.5 = -1.258305739

Reorder the terms:
1.5 + n = -1.258305739

Solving
1.5 + n = -1.258305739

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 = -1.258305739 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = -1.258305739 + -1.5
n = -1.258305739 + -1.5

Combine like terms: -1.258305739 + -1.5 = -2.758305739
n = -2.758305739

Simplifying
n = -2.758305739

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.241694261, -2.758305739}
Solution
n = {-0.241694261, -2.758305739}