2n^2+5n+1=0

Solution for 2n^2+5n+1=0 equation:

Simplifying
2n2 + 5n + 1 = 0

Reorder the terms:
1 + 5n + 2n2 = 0

Solving
1 + 5n + 2n2 = 0

Solving for variable 'n'.

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

Divide each side by '2'.
0.5 + 2.5n + n2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.5n + -0.5 + n2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.5n + n2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.5n + n2 = 0 + -0.5
2.5n + n2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.5n + n2 = -0.5

The n term is 2.5n.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5n + 1.5625 + n2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + 2.5n + n2 = 1.0625

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(n + 1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
n + 1.25 = 1.030776406

Simplifying
n + 1.25 = 1.030776406

Reorder the terms:
1.25 + n = 1.030776406

Solving
1.25 + n = 1.030776406

Solving for variable 'n'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + n = 1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + n = 1.030776406 + -1.25
n = 1.030776406 + -1.25

Combine like terms: 1.030776406 + -1.25 = -0.219223594
n = -0.219223594

Simplifying
n = -0.219223594

Subproblem 2
n + 1.25 = -1.030776406

Simplifying
n + 1.25 = -1.030776406

Reorder the terms:
1.25 + n = -1.030776406

Solving
1.25 + n = -1.030776406

Solving for variable 'n'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + n = -1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + n = -1.030776406 + -1.25
n = -1.030776406 + -1.25

Combine like terms: -1.030776406 + -1.25 = -2.280776406
n = -2.280776406

Simplifying
n = -2.280776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.219223594, -2.280776406}