4n^2=-8n-1

Solution for 4n^2=-8n-1 equation:

Simplifying
4n2 = -8n + -1

Reorder the terms:
4n2 = -1 + -8n

Solving
4n2 = -1 + -8n

Solving for variable 'n'.

Reorder the terms:
1 + 8n + 4n2 = -1 + -8n + 1 + 8n

Reorder the terms:
1 + 8n + 4n2 = -1 + 1 + -8n + 8n

Combine like terms: -1 + 1 = 0
1 + 8n + 4n2 = 0 + -8n + 8n
1 + 8n + 4n2 = -8n + 8n

Combine like terms: -8n + 8n = 0
1 + 8n + 4n2 = 0

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

Divide each side by '4'.
0.25 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 2n + -0.25 + n2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 2n + n2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 2n + n2 = 0 + -0.25
2n + n2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
2n + n2 = -0.25

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 = -0.25 + 1

Reorder the terms:
1 + 2n + n2 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + 2n + n2 = 0.75

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(n + 1) equal to 0.866025404 and -0.866025404.

Subproblem 1
n + 1 = 0.866025404

Simplifying
n + 1 = 0.866025404

Reorder the terms:
1 + n = 0.866025404

Solving
1 + n = 0.866025404

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 = 0.866025404 + -1

Combine like terms: 1 + -1 = 0
0 + n = 0.866025404 + -1
n = 0.866025404 + -1

Combine like terms: 0.866025404 + -1 = -0.133974596
n = -0.133974596

Simplifying
n = -0.133974596

Subproblem 2
n + 1 = -0.866025404

Simplifying
n + 1 = -0.866025404

Reorder the terms:
1 + n = -0.866025404

Solving
1 + n = -0.866025404

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 = -0.866025404 + -1

Combine like terms: 1 + -1 = 0
0 + n = -0.866025404 + -1
n = -0.866025404 + -1

Combine like terms: -0.866025404 + -1 = -1.866025404
n = -1.866025404

Simplifying
n = -1.866025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.133974596, -1.866025404}