1+4n+2n^2=0

Solution for 1+4n+2n^2=0 equation:

Simplifying
1 + 4n + 2n2 = 0

Solving
1 + 4n + 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 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2n + -0.5 + n2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2n + n2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2n + n2 = 0 + -0.5
2n + n2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2n + n2 = -0.5

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 = -0.5 + 1

Reorder the terms:
1 + 2n + n2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2n + n2 = 0.5

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(n + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
n + 1 = 0.707106781

Simplifying
n + 1 = 0.707106781

Reorder the terms:
1 + n = 0.707106781

Solving
1 + n = 0.707106781

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 = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0.707106781 + -1
n = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
n = -0.292893219

Simplifying
n = -0.292893219

Subproblem 2
n + 1 = -0.707106781

Simplifying
n + 1 = -0.707106781

Reorder the terms:
1 + n = -0.707106781

Solving
1 + n = -0.707106781

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 = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + n = -0.707106781 + -1
n = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
n = -1.707106781

Simplifying
n = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.292893219, -1.707106781}