n^2+3*n+1=0

Solution for n^2+3*n+1=0 equation:

Simplifying
n2 + 3n + 1 = 0

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

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

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

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

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

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = -1 + 2.25

Reorder the terms:
2.25 + 3n + n2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3n + n2 = 1.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(n + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
n + 1.5 = 1.118033989

Simplifying
n + 1.5 = 1.118033989

Reorder the terms:
1.5 + n = 1.118033989

Solving
1.5 + n = 1.118033989

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = 1.118033989 + -1.5
n = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
n = -0.381966011

Simplifying
n = -0.381966011

Subproblem 2
n + 1.5 = -1.118033989

Simplifying
n + 1.5 = -1.118033989

Reorder the terms:
1.5 + n = -1.118033989

Solving
1.5 + n = -1.118033989

Solving for variable 'n'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + n = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + n = -1.118033989 + -1.5
n = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
n = -2.618033989

Simplifying
n = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.381966011, -2.618033989}