90(n+18)(2n)=180

Solution for 90(n+18)(2n)=180 equation:

Simplifying
90(n + 18)(2n) = 180

Reorder the terms:
90(18 + n)(2n) = 180

Remove parenthesis around (2n)
90(18 + n) * 2n = 180

Reorder the terms for easier multiplication:
90 * 2n(18 + n) = 180

Multiply 90 * 2
180n(18 + n) = 180
(18 * 180n + n * 180n) = 180
(3240n + 180n2) = 180

Solving
3240n + 180n2 = 180

Solving for variable 'n'.

Reorder the terms:
-180 + 3240n + 180n2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 3240n + 180n2 = 0

Factor out the Greatest Common Factor (GCF), '180'.
180(-1 + 18n + n2) = 0

Ignore the factor 180.
Subproblem 1
Set the factor '(-1 + 18n + n2)' equal to zero and attempt to solve:

Simplifying
-1 + 18n + n2 = 0

Solving
-1 + 18n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 18n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 18n + n2 = 0 + 1

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

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

The n term is 18n.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18n + 81 + n2 = 1 + 81

Reorder the terms:
81 + 18n + n2 = 1 + 81

Combine like terms: 1 + 81 = 82
81 + 18n + n2 = 82

Factor a perfect square on the left side:
(n + 9)(n + 9) = 82

Calculate the square root of the right side: 9.055385138

Break this problem into two subproblems by setting 
(n + 9) equal to 9.055385138 and -9.055385138.

Subproblem 1
n + 9 = 9.055385138

Simplifying
n + 9 = 9.055385138

Reorder the terms:
9 + n = 9.055385138

Solving
9 + n = 9.055385138

Solving for variable 'n'.

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

Add '-9' to each side of the equation.
9 + -9 + n = 9.055385138 + -9

Combine like terms: 9 + -9 = 0
0 + n = 9.055385138 + -9
n = 9.055385138 + -9

Combine like terms: 9.055385138 + -9 = 0.055385138
n = 0.055385138

Simplifying
n = 0.055385138

Subproblem 2
n + 9 = -9.055385138

Simplifying
n + 9 = -9.055385138

Reorder the terms:
9 + n = -9.055385138

Solving
9 + n = -9.055385138

Solving for variable 'n'.

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

Add '-9' to each side of the equation.
9 + -9 + n = -9.055385138 + -9

Combine like terms: 9 + -9 = 0
0 + n = -9.055385138 + -9
n = -9.055385138 + -9

Combine like terms: -9.055385138 + -9 = -18.055385138
n = -18.055385138

Simplifying
n = -18.055385138

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.055385138, -18.055385138}
Solution
n = {0.055385138, -18.055385138}