3n(n-9)=-18

Solution for 3n(n-9)=-18 equation:

Simplifying
3n(n + -9) = -18

Reorder the terms:
3n(-9 + n) = -18
(-9 * 3n + n * 3n) = -18
(-27n + 3n2) = -18

Solving
-27n + 3n2 = -18

Solving for variable 'n'.

Reorder the terms:
18 + -27n + 3n2 = -18 + 18

Combine like terms: -18 + 18 = 0
18 + -27n + 3n2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(6 + -9n + n2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(6 + -9n + n2)' equal to zero and attempt to solve:

Simplifying
6 + -9n + n2 = 0

Solving
6 + -9n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -9n + -6 + n2 = 0 + -6

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

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

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

The n term is -9n.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9n + 20.25 + n2 = -6 + 20.25

Reorder the terms:
20.25 + -9n + n2 = -6 + 20.25

Combine like terms: -6 + 20.25 = 14.25
20.25 + -9n + n2 = 14.25

Factor a perfect square on the left side:
(n + -4.5)(n + -4.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + -4.5) equal to 3.774917218 and -3.774917218.

Subproblem 1
n + -4.5 = 3.774917218

Simplifying
n + -4.5 = 3.774917218

Reorder the terms:
-4.5 + n = 3.774917218

Solving
-4.5 + n = 3.774917218

Solving for variable 'n'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + n = 3.774917218 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + n = 3.774917218 + 4.5
n = 3.774917218 + 4.5

Combine like terms: 3.774917218 + 4.5 = 8.274917218
n = 8.274917218

Simplifying
n = 8.274917218

Subproblem 2
n + -4.5 = -3.774917218

Simplifying
n + -4.5 = -3.774917218

Reorder the terms:
-4.5 + n = -3.774917218

Solving
-4.5 + n = -3.774917218

Solving for variable 'n'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + n = -3.774917218 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + n = -3.774917218 + 4.5
n = -3.774917218 + 4.5

Combine like terms: -3.774917218 + 4.5 = 0.725082782
n = 0.725082782

Simplifying
n = 0.725082782

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.274917218, 0.725082782}
Solution
n = {8.274917218, 0.725082782}