3n^2=-6n-1

Solution for 3n^2=-6n-1 equation:

Simplifying
3n2 = -6n + -1

Reorder the terms:
3n2 = -1 + -6n

Solving
3n2 = -1 + -6n

Solving for variable 'n'.

Reorder the terms:
1 + 6n + 3n2 = -1 + -6n + 1 + 6n

Reorder the terms:
1 + 6n + 3n2 = -1 + 1 + -6n + 6n

Combine like terms: -1 + 1 = 0
1 + 6n + 3n2 = 0 + -6n + 6n
1 + 6n + 3n2 = -6n + 6n

Combine like terms: -6n + 6n = 0
1 + 6n + 3n2 = 0

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

Divide each side by '3'.
0.3333333333 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + 2n + -0.3333333333 + n2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + 2n + n2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + 2n + n2 = 0 + -0.3333333333
2n + n2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
2n + n2 = -0.3333333333

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = -0.3333333333 + 1

Reorder the terms:
1 + 2n + n2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + 2n + n2 = 0.6666666667

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(n + 1) equal to 0.816496581 and -0.816496581.

Subproblem 1
n + 1 = 0.816496581

Simplifying
n + 1 = 0.816496581

Reorder the terms:
1 + n = 0.816496581

Solving
1 + n = 0.816496581

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = 0.816496581 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0.816496581 + -1
n = 0.816496581 + -1

Combine like terms: 0.816496581 + -1 = -0.183503419
n = -0.183503419

Simplifying
n = -0.183503419

Subproblem 2
n + 1 = -0.816496581

Simplifying
n + 1 = -0.816496581

Reorder the terms:
1 + n = -0.816496581

Solving
1 + n = -0.816496581

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = -0.816496581 + -1

Combine like terms: 1 + -1 = 0
0 + n = -0.816496581 + -1
n = -0.816496581 + -1

Combine like terms: -0.816496581 + -1 = -1.816496581
n = -1.816496581

Simplifying
n = -1.816496581

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.183503419, -1.816496581}