6=0.5n(n-1)

Solution for 6=0.5n(n-1) equation:

Simplifying
6 = 0.5n(n + -1)

Reorder the terms:
6 = 0.5n(-1 + n)
6 = (-1 * 0.5n + n * 0.5n)
6 = (-0.5n + 0.5n2)

Solving
6 = -0.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
6 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2

Combine like terms: -0.5n + 0.5n = 0.0
6 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
6 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
6 + 0.5n + -0.5n2 = 0.0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-12 + -1n + n2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -1n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + -1n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -1n + n2 = 0 + 12
-1n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
-1n + n2 = 12

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 12 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 12 + 0.25

Combine like terms: 12 + 0.25 = 12.25
0.25 + -1n + n2 = 12.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 12.25

Calculate the square root of the right side: 3.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 3.5 and -3.5.

Subproblem 1
n + -0.5 = 3.5

Simplifying
n + -0.5 = 3.5

Reorder the terms:
-0.5 + n = 3.5

Solving
-0.5 + n = 3.5

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = 3.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 3.5 + 0.5
n = 3.5 + 0.5

Combine like terms: 3.5 + 0.5 = 4
n = 4

Simplifying
n = 4

Subproblem 2
n + -0.5 = -3.5

Simplifying
n + -0.5 = -3.5

Reorder the terms:
-0.5 + n = -3.5

Solving
-0.5 + n = -3.5

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = -3.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -3.5 + 0.5
n = -3.5 + 0.5

Combine like terms: -3.5 + 0.5 = -3
n = -3

Simplifying
n = -3

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4, -3}