0.5*n(n-3)=54

Solution for 0.5*n(n-3)=54 equation:

Simplifying
0.5n(n + -3) = 54

Reorder the terms:
0.5n(-3 + n) = 54
(-3 * 0.5n + n * 0.5n) = 54
(-1.5n + 0.5n2) = 54

Solving
-1.5n + 0.5n2 = 54

Solving for variable 'n'.

Reorder the terms:
-54 + -1.5n + 0.5n2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + -1.5n + 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'.
-108 + -3n + n2 = 0

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + -3n + 108 + n2 = 0 + 108

Reorder the terms:
-108 + 108 + -3n + n2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + -3n + n2 = 0 + 108
-3n + n2 = 0 + 108

Combine like terms: 0 + 108 = 108
-3n + n2 = 108

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 108 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 108 + 2.25

Combine like terms: 108 + 2.25 = 110.25
2.25 + -3n + n2 = 110.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 110.25

Calculate the square root of the right side: 10.5

Break this problem into two subproblems by setting 
(n + -1.5) equal to 10.5 and -10.5.

Subproblem 1
n + -1.5 = 10.5

Simplifying
n + -1.5 = 10.5

Reorder the terms:
-1.5 + n = 10.5

Solving
-1.5 + n = 10.5

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = 10.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 10.5 + 1.5
n = 10.5 + 1.5

Combine like terms: 10.5 + 1.5 = 12
n = 12

Simplifying
n = 12

Subproblem 2
n + -1.5 = -10.5

Simplifying
n + -1.5 = -10.5

Reorder the terms:
-1.5 + n = -10.5

Solving
-1.5 + n = -10.5

Solving for variable 'n'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + n = -10.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -10.5 + 1.5
n = -10.5 + 1.5

Combine like terms: -10.5 + 1.5 = -9
n = -9

Simplifying
n = -9

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {12, -9}