0.5n^2+0.5n=120

Solution for 0.5n^2+0.5n=120 equation:

Simplifying
0.5n2 + 0.5n = 120

Reorder the terms:
0.5n + 0.5n2 = 120

Solving
0.5n + 0.5n2 = 120

Solving for variable 'n'.

Reorder the terms:
-120 + 0.5n + 0.5n2 = 120 + -120

Combine like terms: 120 + -120 = 0
-120 + 0.5n + 0.5n2 = 0

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

Divide each side by '0.5'.
-240 + n + n2 = 0

Move the constant term to the right:

Add '240' to each side of the equation.
-240 + n + 240 + n2 = 0 + 240

Reorder the terms:
-240 + 240 + n + n2 = 0 + 240

Combine like terms: -240 + 240 = 0
0 + n + n2 = 0 + 240
n + n2 = 0 + 240

Combine like terms: 0 + 240 = 240
n + n2 = 240

The n term is n.  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.
 + 0.25 + n2 = 240 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 240 + 0.25

Combine like terms: 240 + 0.25 = 240.25
1.25 + n2 = 240.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 240.25

Calculate the square root of the right side: 15.5

Break this problem into two subproblems by setting 
(n + 0.5) equal to 15.5 and -15.5.

Subproblem 1
n + 0.5 = 15.5

Simplifying
n + 0.5 = 15.5

Reorder the terms:
0.5 + n = 15.5

Solving
0.5 + n = 15.5

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 = 15.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 15.5 + -0.5
n = 15.5 + -0.5

Combine like terms: 15.5 + -0.5 = 15
n = 15

Simplifying
n = 15

Subproblem 2
n + 0.5 = -15.5

Simplifying
n + 0.5 = -15.5

Reorder the terms:
0.5 + n = -15.5

Solving
0.5 + n = -15.5

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 = -15.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -15.5 + -0.5
n = -15.5 + -0.5

Combine like terms: -15.5 + -0.5 = -16
n = -16

Simplifying
n = -16

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {15, -16}