9n^2+3n-15=0

Solution for 9n^2+3n-15=0 equation:

Simplifying
9n2 + 3n + -15 = 0

Reorder the terms:
-15 + 3n + 9n2 = 0

Solving
-15 + 3n + 9n2 = 0

Solving for variable 'n'.

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

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

Simplifying
-5 + n + 3n2 = 0

Solving
-5 + n + 3n2 = 0

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

Divide each side by '3'.
-1.666666667 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 0.3333333333n + 1.666666667 + n2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 0.3333333333n + n2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 0.3333333333n + n2 = 0 + 1.666666667
0.3333333333n + n2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
0.3333333333n + n2 = 1.666666667

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.3333333333n + 0.25 + n2 = 1.666666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 1.666666667 + 0.25

Combine like terms: 1.666666667 + 0.25 = 1.916666667
0.25 + 0.3333333333n + n2 = 1.916666667

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1.916666667

Calculate the square root of the right side: 1.384437311

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.384437311 and -1.384437311.

Subproblem 1
n + 0.5 = 1.384437311

Simplifying
n + 0.5 = 1.384437311

Reorder the terms:
0.5 + n = 1.384437311

Solving
0.5 + n = 1.384437311

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 1.384437311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 1.384437311 + -0.5
n = 1.384437311 + -0.5

Combine like terms: 1.384437311 + -0.5 = 0.884437311
n = 0.884437311

Simplifying
n = 0.884437311

Subproblem 2
n + 0.5 = -1.384437311

Simplifying
n + 0.5 = -1.384437311

Reorder the terms:
0.5 + n = -1.384437311

Solving
0.5 + n = -1.384437311

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -1.384437311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -1.384437311 + -0.5
n = -1.384437311 + -0.5

Combine like terms: -1.384437311 + -0.5 = -1.884437311
n = -1.884437311

Simplifying
n = -1.884437311

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.884437311, -1.884437311}
Solution
n = {0.884437311, -1.884437311}