n^2+6n+1=0

Solution for n^2+6n+1=0 equation:

Simplifying
n2 + 6n + 1 = 0

Reorder the terms:
1 + 6n + n2 = 0

Solving
1 + 6n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 6n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 6n + n2 = 0 + -1

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

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

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = -1 + 9

Reorder the terms:
9 + 6n + n2 = -1 + 9

Combine like terms: -1 + 9 = 8
9 + 6n + n2 = 8

Factor a perfect square on the left side:
(n + 3)(n + 3) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(n + 3) equal to 2.828427125 and -2.828427125.

Subproblem 1
n + 3 = 2.828427125

Simplifying
n + 3 = 2.828427125

Reorder the terms:
3 + n = 2.828427125

Solving
3 + n = 2.828427125

Solving for variable 'n'.

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

Add '-3' to each side of the equation.
3 + -3 + n = 2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + n = 2.828427125 + -3
n = 2.828427125 + -3

Combine like terms: 2.828427125 + -3 = -0.171572875
n = -0.171572875

Simplifying
n = -0.171572875

Subproblem 2
n + 3 = -2.828427125

Simplifying
n + 3 = -2.828427125

Reorder the terms:
3 + n = -2.828427125

Solving
3 + n = -2.828427125

Solving for variable 'n'.

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

Add '-3' to each side of the equation.
3 + -3 + n = -2.828427125 + -3

Combine like terms: 3 + -3 = 0
0 + n = -2.828427125 + -3
n = -2.828427125 + -3

Combine like terms: -2.828427125 + -3 = -5.828427125
n = -5.828427125

Simplifying
n = -5.828427125

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.171572875, -5.828427125}