5n(3n+1)=15

Solution for 5n(3n+1)=15 equation:

Simplifying
5n(3n + 1) = 15

Reorder the terms:
5n(1 + 3n) = 15
(1 * 5n + 3n * 5n) = 15
(5n + 15n2) = 15

Solving
5n + 15n2 = 15

Solving for variable 'n'.

Reorder the terms:
-15 + 5n + 15n2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 5n + 15n2 = 0

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

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

Simplifying
-3 + n + 3n2 = 0

Solving
-3 + n + 3n2 = 0

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

Divide each side by '3'.
-1 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.3333333333n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.3333333333n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.3333333333n + n2 = 0 + 1
0.3333333333n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.3333333333n + n2 = 1

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 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 1 + 0.25

Combine like terms: 1 + 0.25 = 1.25
0.25 + 0.3333333333n + n2 = 1.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
n + 0.5 = 1.118033989

Simplifying
n + 0.5 = 1.118033989

Reorder the terms:
0.5 + n = 1.118033989

Solving
0.5 + n = 1.118033989

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 = 1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 1.118033989 + -0.5
n = 1.118033989 + -0.5

Combine like terms: 1.118033989 + -0.5 = 0.618033989
n = 0.618033989

Simplifying
n = 0.618033989

Subproblem 2
n + 0.5 = -1.118033989

Simplifying
n + 0.5 = -1.118033989

Reorder the terms:
0.5 + n = -1.118033989

Solving
0.5 + n = -1.118033989

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 = -1.118033989 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -1.118033989 + -0.5
n = -1.118033989 + -0.5

Combine like terms: -1.118033989 + -0.5 = -1.618033989
n = -1.618033989

Simplifying
n = -1.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.618033989, -1.618033989}
Solution
n = {0.618033989, -1.618033989}