5n^2-2n-6=0

Solution for 5n^2-2n-6=0 equation:

Simplifying
5n2 + -2n + -6 = 0

Reorder the terms:
-6 + -2n + 5n2 = 0

Solving
-6 + -2n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
-1.2 + -0.4n + n2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + -0.4n + 1.2 + n2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + -0.4n + n2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + -0.4n + n2 = 0 + 1.2
-0.4n + n2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
-0.4n + n2 = 1.2

The n term is -0.4n.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4n + 0.04 + n2 = 1.2 + 0.04

Reorder the terms:
0.04 + -0.4n + n2 = 1.2 + 0.04

Combine like terms: 1.2 + 0.04 = 1.24
0.04 + -0.4n + n2 = 1.24

Factor a perfect square on the left side:
(n + -0.2)(n + -0.2) = 1.24

Calculate the square root of the right side: 1.113552873

Break this problem into two subproblems by setting 
(n + -0.2) equal to 1.113552873 and -1.113552873.

Subproblem 1
n + -0.2 = 1.113552873

Simplifying
n + -0.2 = 1.113552873

Reorder the terms:
-0.2 + n = 1.113552873

Solving
-0.2 + n = 1.113552873

Solving for variable 'n'.

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

Add '0.2' to each side of the equation.
-0.2 + 0.2 + n = 1.113552873 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + n = 1.113552873 + 0.2
n = 1.113552873 + 0.2

Combine like terms: 1.113552873 + 0.2 = 1.313552873
n = 1.313552873

Simplifying
n = 1.313552873

Subproblem 2
n + -0.2 = -1.113552873

Simplifying
n + -0.2 = -1.113552873

Reorder the terms:
-0.2 + n = -1.113552873

Solving
-0.2 + n = -1.113552873

Solving for variable 'n'.

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

Add '0.2' to each side of the equation.
-0.2 + 0.2 + n = -1.113552873 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + n = -1.113552873 + 0.2
n = -1.113552873 + 0.2

Combine like terms: -1.113552873 + 0.2 = -0.913552873
n = -0.913552873

Simplifying
n = -0.913552873

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.313552873, -0.913552873}