105=0.5(n^2-n)

Solution for 105=0.5(n^2-n) equation:

Simplifying
105 = 0.5(n2 + -1n)

Reorder the terms:
105 = 0.5(-1n + n2)
105 = (-1n * 0.5 + n2 * 0.5)
105 = (-0.5n + 0.5n2)

Solving
105 = -0.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
105 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2

Combine like terms: -0.5n + 0.5n = 0.0
105 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
105 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
105 + 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'.
-210 + -1n + n2 = 0

Move the constant term to the right:

Add '210' to each side of the equation.
-210 + -1n + 210 + n2 = 0 + 210

Reorder the terms:
-210 + 210 + -1n + n2 = 0 + 210

Combine like terms: -210 + 210 = 0
0 + -1n + n2 = 0 + 210
-1n + n2 = 0 + 210

Combine like terms: 0 + 210 = 210
-1n + n2 = 210

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 = 210 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 210 + 0.25

Combine like terms: 210 + 0.25 = 210.25
0.25 + -1n + n2 = 210.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 210.25

Calculate the square root of the right side: 14.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 14.5 and -14.5.

Subproblem 1
n + -0.5 = 14.5

Simplifying
n + -0.5 = 14.5

Reorder the terms:
-0.5 + n = 14.5

Solving
-0.5 + n = 14.5

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 = 14.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 14.5 + 0.5
n = 14.5 + 0.5

Combine like terms: 14.5 + 0.5 = 15
n = 15

Simplifying
n = 15

Subproblem 2
n + -0.5 = -14.5

Simplifying
n + -0.5 = -14.5

Reorder the terms:
-0.5 + n = -14.5

Solving
-0.5 + n = -14.5

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 = -14.5 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -14.5 + 0.5
n = -14.5 + 0.5

Combine like terms: -14.5 + 0.5 = -14
n = -14

Simplifying
n = -14

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {15, -14}