3n(3n-18)-(n-6)(n-1)=40(n-1)

Solution for 3n(3n-18)-(n-6)(n-1)=40(n-1) equation:

Simplifying
3n(3n + -18) + -1(n + -6)(n + -1) = 40(n + -1)

Reorder the terms:
3n(-18 + 3n) + -1(n + -6)(n + -1) = 40(n + -1)
(-18 * 3n + 3n * 3n) + -1(n + -6)(n + -1) = 40(n + -1)
(-54n + 9n2) + -1(n + -6)(n + -1) = 40(n + -1)

Reorder the terms:
-54n + 9n2 + -1(-6 + n)(n + -1) = 40(n + -1)

Reorder the terms:
-54n + 9n2 + -1(-6 + n)(-1 + n) = 40(n + -1)

Multiply (-6 + n) * (-1 + n)
-54n + 9n2 + -1(-6(-1 + n) + n(-1 + n)) = 40(n + -1)
-54n + 9n2 + -1((-1 * -6 + n * -6) + n(-1 + n)) = 40(n + -1)
-54n + 9n2 + -1((6 + -6n) + n(-1 + n)) = 40(n + -1)
-54n + 9n2 + -1(6 + -6n + (-1 * n + n * n)) = 40(n + -1)
-54n + 9n2 + -1(6 + -6n + (-1n + n2)) = 40(n + -1)

Combine like terms: -6n + -1n = -7n
-54n + 9n2 + -1(6 + -7n + n2) = 40(n + -1)
-54n + 9n2 + (6 * -1 + -7n * -1 + n2 * -1) = 40(n + -1)
-54n + 9n2 + (-6 + 7n + -1n2) = 40(n + -1)

Reorder the terms:
-6 + -54n + 7n + 9n2 + -1n2 = 40(n + -1)

Combine like terms: -54n + 7n = -47n
-6 + -47n + 9n2 + -1n2 = 40(n + -1)

Combine like terms: 9n2 + -1n2 = 8n2
-6 + -47n + 8n2 = 40(n + -1)

Reorder the terms:
-6 + -47n + 8n2 = 40(-1 + n)
-6 + -47n + 8n2 = (-1 * 40 + n * 40)
-6 + -47n + 8n2 = (-40 + 40n)

Solving
-6 + -47n + 8n2 = -40 + 40n

Solving for variable 'n'.

Reorder the terms:
-6 + 40 + -47n + -40n + 8n2 = -40 + 40n + 40 + -40n

Combine like terms: -6 + 40 = 34
34 + -47n + -40n + 8n2 = -40 + 40n + 40 + -40n

Combine like terms: -47n + -40n = -87n
34 + -87n + 8n2 = -40 + 40n + 40 + -40n

Reorder the terms:
34 + -87n + 8n2 = -40 + 40 + 40n + -40n

Combine like terms: -40 + 40 = 0
34 + -87n + 8n2 = 0 + 40n + -40n
34 + -87n + 8n2 = 40n + -40n

Combine like terms: 40n + -40n = 0
34 + -87n + 8n2 = 0

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

Divide each side by '8'.
4.25 + -10.875n + n2 = 0

Move the constant term to the right:

Add '-4.25' to each side of the equation.
4.25 + -10.875n + -4.25 + n2 = 0 + -4.25

Reorder the terms:
4.25 + -4.25 + -10.875n + n2 = 0 + -4.25

Combine like terms: 4.25 + -4.25 = 0.00
0.00 + -10.875n + n2 = 0 + -4.25
-10.875n + n2 = 0 + -4.25

Combine like terms: 0 + -4.25 = -4.25
-10.875n + n2 = -4.25

The n term is -10.875n.  Take half its coefficient (-5.4375).
Square it (29.56640625) and add it to both sides.

Add '29.56640625' to each side of the equation.
-10.875n + 29.56640625 + n2 = -4.25 + 29.56640625

Reorder the terms:
29.56640625 + -10.875n + n2 = -4.25 + 29.56640625

Combine like terms: -4.25 + 29.56640625 = 25.31640625
29.56640625 + -10.875n + n2 = 25.31640625

Factor a perfect square on the left side:
(n + -5.4375)(n + -5.4375) = 25.31640625

Calculate the square root of the right side: 5.031541141

Break this problem into two subproblems by setting 
(n + -5.4375) equal to 5.031541141 and -5.031541141.

Subproblem 1
n + -5.4375 = 5.031541141

Simplifying
n + -5.4375 = 5.031541141

Reorder the terms:
-5.4375 + n = 5.031541141

Solving
-5.4375 + n = 5.031541141

Solving for variable 'n'.

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

Add '5.4375' to each side of the equation.
-5.4375 + 5.4375 + n = 5.031541141 + 5.4375

Combine like terms: -5.4375 + 5.4375 = 0.0000
0.0000 + n = 5.031541141 + 5.4375
n = 5.031541141 + 5.4375

Combine like terms: 5.031541141 + 5.4375 = 10.469041141
n = 10.469041141

Simplifying
n = 10.469041141

Subproblem 2
n + -5.4375 = -5.031541141

Simplifying
n + -5.4375 = -5.031541141

Reorder the terms:
-5.4375 + n = -5.031541141

Solving
-5.4375 + n = -5.031541141

Solving for variable 'n'.

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

Add '5.4375' to each side of the equation.
-5.4375 + 5.4375 + n = -5.031541141 + 5.4375

Combine like terms: -5.4375 + 5.4375 = 0.0000
0.0000 + n = -5.031541141 + 5.4375
n = -5.031541141 + 5.4375

Combine like terms: -5.031541141 + 5.4375 = 0.405958859
n = 0.405958859

Simplifying
n = 0.405958859

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.469041141, 0.405958859}