120=.5l^2+.5l

Solution for 120=.5l^2+.5l equation:

Simplifying
120 = 0.5l2 + 0.5l

Reorder the terms:
120 = 0.5l + 0.5l2

Solving
120 = 0.5l + 0.5l2

Solving for variable 'l'.

Reorder the terms:
120 + -0.5l + -0.5l2 = 0.5l + -0.5l + 0.5l2 + -0.5l2

Combine like terms: 0.5l + -0.5l = 0.0
120 + -0.5l + -0.5l2 = 0.0 + 0.5l2 + -0.5l2
120 + -0.5l + -0.5l2 = 0.5l2 + -0.5l2

Combine like terms: 0.5l2 + -0.5l2 = 0.0
120 + -0.5l + -0.5l2 = 0.0

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

Divide each side by '-0.5'.
-240 + l + l2 = 0

Move the constant term to the right:

Add '240' to each side of the equation.
-240 + l + 240 + l2 = 0 + 240

Reorder the terms:
-240 + 240 + l + l2 = 0 + 240

Combine like terms: -240 + 240 = 0
0 + l + l2 = 0 + 240
l + l2 = 0 + 240

Combine like terms: 0 + 240 = 240
l + l2 = 240

The l term is l.  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 + l2 = 240 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + l2 = 240 + 0.25

Combine like terms: 240 + 0.25 = 240.25
1.25 + l2 = 240.25

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

Calculate the square root of the right side: 15.5

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

Subproblem 1
l + 0.5 = 15.5

Simplifying
l + 0.5 = 15.5

Reorder the terms:
0.5 + l = 15.5

Solving
0.5 + l = 15.5

Solving for variable 'l'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + l = 15.5 + -0.5

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

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

Simplifying
l = 15

Subproblem 2
l + 0.5 = -15.5

Simplifying
l + 0.5 = -15.5

Reorder the terms:
0.5 + l = -15.5

Solving
0.5 + l = -15.5

Solving for variable 'l'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + l = -15.5 + -0.5

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

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

Simplifying
l = -16

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