90=0.5n(n-3)

Solution for 90=0.5n(n-3) equation:

Simplifying
90 = 0.5n(n + -3)

Reorder the terms:
90 = 0.5n(-3 + n)
90 = (-3 * 0.5n + n * 0.5n)
90 = (-1.5n + 0.5n2)

Solving
90 = -1.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
90 + 1.5n + -0.5n2 = -1.5n + 1.5n + 0.5n2 + -0.5n2

Combine like terms: -1.5n + 1.5n = 0.0
90 + 1.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
90 + 1.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
90 + 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'.
-180 + -3n + n2 = 0

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + -3n + 180 + n2 = 0 + 180

Reorder the terms:
-180 + 180 + -3n + n2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + -3n + n2 = 0 + 180
-3n + n2 = 0 + 180

Combine like terms: 0 + 180 = 180
-3n + n2 = 180

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 = 180 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 180 + 2.25

Combine like terms: 180 + 2.25 = 182.25
2.25 + -3n + n2 = 182.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 182.25

Calculate the square root of the right side: 13.5

Break this problem into two subproblems by setting 
(n + -1.5) equal to 13.5 and -13.5.

Subproblem 1
n + -1.5 = 13.5

Simplifying
n + -1.5 = 13.5

Reorder the terms:
-1.5 + n = 13.5

Solving
-1.5 + n = 13.5

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 = 13.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 13.5 + 1.5
n = 13.5 + 1.5

Combine like terms: 13.5 + 1.5 = 15
n = 15

Simplifying
n = 15

Subproblem 2
n + -1.5 = -13.5

Simplifying
n + -1.5 = -13.5

Reorder the terms:
-1.5 + n = -13.5

Solving
-1.5 + n = -13.5

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 = -13.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -13.5 + 1.5
n = -13.5 + 1.5

Combine like terms: -13.5 + 1.5 = -12
n = -12

Simplifying
n = -12

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {15, -12}