0=3n^2-n-111

Solution for 0=3n^2-n-111 equation:

Simplifying
0 = 3n2 + -1n + -111

Reorder the terms:
0 = -111 + -1n + 3n2

Solving
0 = -111 + -1n + 3n2

Solving for variable 'n'.

Combine like terms: 0 + 111 = 111
111 + n + -3n2 = -111 + -1n + 3n2 + 111 + n + -3n2

Reorder the terms:
111 + n + -3n2 = -111 + 111 + -1n + n + 3n2 + -3n2

Combine like terms: -111 + 111 = 0
111 + n + -3n2 = 0 + -1n + n + 3n2 + -3n2
111 + n + -3n2 = -1n + n + 3n2 + -3n2

Combine like terms: -1n + n = 0
111 + n + -3n2 = 0 + 3n2 + -3n2
111 + n + -3n2 = 3n2 + -3n2

Combine like terms: 3n2 + -3n2 = 0
111 + n + -3n2 = 0

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

Divide each side by '-3'.
-37 + -0.3333333333n + n2 = 0

Move the constant term to the right:

Add '37' to each side of the equation.
-37 + -0.3333333333n + 37 + n2 = 0 + 37

Reorder the terms:
-37 + 37 + -0.3333333333n + n2 = 0 + 37

Combine like terms: -37 + 37 = 0
0 + -0.3333333333n + n2 = 0 + 37
-0.3333333333n + n2 = 0 + 37

Combine like terms: 0 + 37 = 37
-0.3333333333n + n2 = 37

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 = 37 + 0.25

Reorder the terms:
0.25 + -0.3333333333n + n2 = 37 + 0.25

Combine like terms: 37 + 0.25 = 37.25
0.25 + -0.3333333333n + n2 = 37.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 37.25

Calculate the square root of the right side: 6.103277808

Break this problem into two subproblems by setting 
(n + 0.5) equal to 6.103277808 and -6.103277808.

Subproblem 1
n + 0.5 = 6.103277808

Simplifying
n + 0.5 = 6.103277808

Reorder the terms:
0.5 + n = 6.103277808

Solving
0.5 + n = 6.103277808

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 = 6.103277808 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 6.103277808 + -0.5
n = 6.103277808 + -0.5

Combine like terms: 6.103277808 + -0.5 = 5.603277808
n = 5.603277808

Simplifying
n = 5.603277808

Subproblem 2
n + 0.5 = -6.103277808

Simplifying
n + 0.5 = -6.103277808

Reorder the terms:
0.5 + n = -6.103277808

Solving
0.5 + n = -6.103277808

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 = -6.103277808 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -6.103277808 + -0.5
n = -6.103277808 + -0.5

Combine like terms: -6.103277808 + -0.5 = -6.603277808
n = -6.603277808

Simplifying
n = -6.603277808

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.603277808, -6.603277808}