54=.5N(N-3)

Solution for 54=.5N(N-3) equation:

Simplifying
54 = 0.5N(N + -3)

Reorder the terms:
54 = 0.5N(-3 + N)
54 = (-3 * 0.5N + N * 0.5N)
54 = (-1.5N + 0.5N2)

Solving
54 = -1.5N + 0.5N2

Solving for variable 'N'.

Reorder the terms:
54 + 1.5N + -0.5N2 = -1.5N + 1.5N + 0.5N2 + -0.5N2

Combine like terms: -1.5N + 1.5N = 0.0
54 + 1.5N + -0.5N2 = 0.0 + 0.5N2 + -0.5N2
54 + 1.5N + -0.5N2 = 0.5N2 + -0.5N2

Combine like terms: 0.5N2 + -0.5N2 = 0.0
54 + 1.5N + -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'.
-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}