15n^2+27n+6=0

Solution for 15n^2+27n+6=0 equation:

Simplifying
15n2 + 27n + 6 = 0

Reorder the terms:
6 + 27n + 15n2 = 0

Solving
6 + 27n + 15n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '3'.
3(2 + 9n + 5n2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(2 + 9n + 5n2)' equal to zero and attempt to solve:

Simplifying
2 + 9n + 5n2 = 0

Solving
2 + 9n + 5n2 = 0

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

Divide each side by '5'.
0.4 + 1.8n + n2 = 0

Move the constant term to the right:

Add '-0.4' to each side of the equation.
0.4 + 1.8n + -0.4 + n2 = 0 + -0.4

Reorder the terms:
0.4 + -0.4 + 1.8n + n2 = 0 + -0.4

Combine like terms: 0.4 + -0.4 = 0.0
0.0 + 1.8n + n2 = 0 + -0.4
1.8n + n2 = 0 + -0.4

Combine like terms: 0 + -0.4 = -0.4
1.8n + n2 = -0.4

The n term is 1.8n.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8n + 0.81 + n2 = -0.4 + 0.81

Reorder the terms:
0.81 + 1.8n + n2 = -0.4 + 0.81

Combine like terms: -0.4 + 0.81 = 0.41
0.81 + 1.8n + n2 = 0.41

Factor a perfect square on the left side:
(n + 0.9)(n + 0.9) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(n + 0.9) equal to 0.640312424 and -0.640312424.

Subproblem 1
n + 0.9 = 0.640312424

Simplifying
n + 0.9 = 0.640312424

Reorder the terms:
0.9 + n = 0.640312424

Solving
0.9 + n = 0.640312424

Solving for variable 'n'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + n = 0.640312424 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + n = 0.640312424 + -0.9
n = 0.640312424 + -0.9

Combine like terms: 0.640312424 + -0.9 = -0.259687576
n = -0.259687576

Simplifying
n = -0.259687576

Subproblem 2
n + 0.9 = -0.640312424

Simplifying
n + 0.9 = -0.640312424

Reorder the terms:
0.9 + n = -0.640312424

Solving
0.9 + n = -0.640312424

Solving for variable 'n'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + n = -0.640312424 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + n = -0.640312424 + -0.9
n = -0.640312424 + -0.9

Combine like terms: -0.640312424 + -0.9 = -1.540312424
n = -1.540312424

Simplifying
n = -1.540312424

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.259687576, -1.540312424}
Solution
n = {-0.259687576, -1.540312424}