2n^2-4n-1=0

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

Simplifying
2n2 + -4n + -1 = 0

Reorder the terms:
-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 = 1.5
1 + -2n + n2 = 1.5

Factor a perfect square on the left side:
(n + -1)(n + -1) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(n + -1) equal to 1.224744871 and -1.224744871.

Subproblem 1
n + -1 = 1.224744871

Simplifying
n + -1 = 1.224744871

Reorder the terms:
-1 + n = 1.224744871

Solving
-1 + n = 1.224744871

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 = 1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + n = 1.224744871 + 1
n = 1.224744871 + 1

Combine like terms: 1.224744871 + 1 = 2.224744871
n = 2.224744871

Simplifying
n = 2.224744871

Subproblem 2
n + -1 = -1.224744871

Simplifying
n + -1 = -1.224744871

Reorder the terms:
-1 + n = -1.224744871

Solving
-1 + n = -1.224744871

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 = -1.224744871 + 1

Combine like terms: -1 + 1 = 0
0 + n = -1.224744871 + 1
n = -1.224744871 + 1

Combine like terms: -1.224744871 + 1 = -0.224744871
n = -0.224744871

Simplifying
n = -0.224744871

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.224744871, -0.224744871}