3n^2+6n+2=0

Solution for 3n^2+6n+2=0 equation:

Simplifying
3n2 + 6n + 2 = 0

Reorder the terms:
2 + 6n + 3n2 = 0

Solving
2 + 6n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
0.6666666667 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + 2n + -0.6666666667 + n2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + 2n + n2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + 2n + n2 = 0 + -0.6666666667
2n + n2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
2n + n2 = -0.6666666667

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 = -0.6666666667 + 1

Reorder the terms:
1 + 2n + n2 = -0.6666666667 + 1

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

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.3333333333

Calculate the square root of the right side: 0.577350269

Break this problem into two subproblems by setting 
(n + 1) equal to 0.577350269 and -0.577350269.

Subproblem 1
n + 1 = 0.577350269

Simplifying
n + 1 = 0.577350269

Reorder the terms:
1 + n = 0.577350269

Solving
1 + n = 0.577350269

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 = 0.577350269 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0.577350269 + -1
n = 0.577350269 + -1

Combine like terms: 0.577350269 + -1 = -0.422649731
n = -0.422649731

Simplifying
n = -0.422649731

Subproblem 2
n + 1 = -0.577350269

Simplifying
n + 1 = -0.577350269

Reorder the terms:
1 + n = -0.577350269

Solving
1 + n = -0.577350269

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 = -0.577350269 + -1

Combine like terms: 1 + -1 = 0
0 + n = -0.577350269 + -1
n = -0.577350269 + -1

Combine like terms: -0.577350269 + -1 = -1.577350269
n = -1.577350269

Simplifying
n = -1.577350269

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.422649731, -1.577350269}