6=6n(3n+33)+7(n+1)

Solution for 6=6n(3n+33)+7(n+1) equation:

Simplifying
6 = 6n(3n + 33) + 7(n + 1)

Reorder the terms:
6 = 6n(33 + 3n) + 7(n + 1)
6 = (33 * 6n + 3n * 6n) + 7(n + 1)
6 = (198n + 18n2) + 7(n + 1)

Reorder the terms:
6 = 198n + 18n2 + 7(1 + n)
6 = 198n + 18n2 + (1 * 7 + n * 7)
6 = 198n + 18n2 + (7 + 7n)

Reorder the terms:
6 = 7 + 198n + 7n + 18n2

Combine like terms: 198n + 7n = 205n
6 = 7 + 205n + 18n2

Solving
6 = 7 + 205n + 18n2

Solving for variable 'n'.

Combine like terms: 6 + -7 = -1
-1 + -205n + -18n2 = 7 + 205n + 18n2 + -7 + -205n + -18n2

Reorder the terms:
-1 + -205n + -18n2 = 7 + -7 + 205n + -205n + 18n2 + -18n2

Combine like terms: 7 + -7 = 0
-1 + -205n + -18n2 = 0 + 205n + -205n + 18n2 + -18n2
-1 + -205n + -18n2 = 205n + -205n + 18n2 + -18n2

Combine like terms: 205n + -205n = 0
-1 + -205n + -18n2 = 0 + 18n2 + -18n2
-1 + -205n + -18n2 = 18n2 + -18n2

Combine like terms: 18n2 + -18n2 = 0
-1 + -205n + -18n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 205n + 18n2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 205n + 18n2)' equal to zero and attempt to solve:

Simplifying
1 + 205n + 18n2 = 0

Solving
1 + 205n + 18n2 = 0

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

Divide each side by '18'.
0.05555555556 + 11.38888889n + n2 = 0

Move the constant term to the right:

Add '-0.05555555556' to each side of the equation.
0.05555555556 + 11.38888889n + -0.05555555556 + n2 = 0 + -0.05555555556

Reorder the terms:
0.05555555556 + -0.05555555556 + 11.38888889n + n2 = 0 + -0.05555555556

Combine like terms: 0.05555555556 + -0.05555555556 = 0.00000000000
0.00000000000 + 11.38888889n + n2 = 0 + -0.05555555556
11.38888889n + n2 = 0 + -0.05555555556

Combine like terms: 0 + -0.05555555556 = -0.05555555556
11.38888889n + n2 = -0.05555555556

The n term is 11.38888889n.  Take half its coefficient (5.694444445).
Square it (32.42669754) and add it to both sides.

Add '32.42669754' to each side of the equation.
11.38888889n + 32.42669754 + n2 = -0.05555555556 + 32.42669754

Reorder the terms:
32.42669754 + 11.38888889n + n2 = -0.05555555556 + 32.42669754

Combine like terms: -0.05555555556 + 32.42669754 = 32.37114198444
32.42669754 + 11.38888889n + n2 = 32.37114198444

Factor a perfect square on the left side:
(n + 5.694444445)(n + 5.694444445) = 32.37114198444

Calculate the square root of the right side: 5.689564305

Break this problem into two subproblems by setting 
(n + 5.694444445) equal to 5.689564305 and -5.689564305.

Subproblem 1
n + 5.694444445 = 5.689564305

Simplifying
n + 5.694444445 = 5.689564305

Reorder the terms:
5.694444445 + n = 5.689564305

Solving
5.694444445 + n = 5.689564305

Solving for variable 'n'.

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

Add '-5.694444445' to each side of the equation.
5.694444445 + -5.694444445 + n = 5.689564305 + -5.694444445

Combine like terms: 5.694444445 + -5.694444445 = 0.000000000
0.000000000 + n = 5.689564305 + -5.694444445
n = 5.689564305 + -5.694444445

Combine like terms: 5.689564305 + -5.694444445 = -0.00488014
n = -0.00488014

Simplifying
n = -0.00488014

Subproblem 2
n + 5.694444445 = -5.689564305

Simplifying
n + 5.694444445 = -5.689564305

Reorder the terms:
5.694444445 + n = -5.689564305

Solving
5.694444445 + n = -5.689564305

Solving for variable 'n'.

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

Add '-5.694444445' to each side of the equation.
5.694444445 + -5.694444445 + n = -5.689564305 + -5.694444445

Combine like terms: 5.694444445 + -5.694444445 = 0.000000000
0.000000000 + n = -5.689564305 + -5.694444445
n = -5.689564305 + -5.694444445

Combine like terms: -5.689564305 + -5.694444445 = -11.38400875
n = -11.38400875

Simplifying
n = -11.38400875

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.00488014, -11.38400875}
Solution
n = {-0.00488014, -11.38400875}