0.5n^2-1.5n-27=0

Solution for 0.5n^2-1.5n-27=0 equation:

Simplifying
0.5n2 + -1.5n + -27 = 0

Reorder the terms:
-27 + -1.5n + 0.5n2 = 0

Solving
-27 + -1.5n + 0.5n2 = 0

Solving for variable 'n'.

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

Divide each side by '0.5'.
-54 + -3n + n2 = 0

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + -3n + 54 + n2 = 0 + 54

Reorder the terms:
-54 + 54 + -3n + n2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + -3n + n2 = 0 + 54
-3n + n2 = 0 + 54

Combine like terms: 0 + 54 = 54
-3n + n2 = 54

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 = 54 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 54 + 2.25

Combine like terms: 54 + 2.25 = 56.25
2.25 + -3n + n2 = 56.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 56.25

Calculate the square root of the right side: 7.5

Break this problem into two subproblems by setting 
(n + -1.5) equal to 7.5 and -7.5.

Subproblem 1
n + -1.5 = 7.5

Simplifying
n + -1.5 = 7.5

Reorder the terms:
-1.5 + n = 7.5

Solving
-1.5 + n = 7.5

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 = 7.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = 7.5 + 1.5
n = 7.5 + 1.5

Combine like terms: 7.5 + 1.5 = 9
n = 9

Simplifying
n = 9

Subproblem 2
n + -1.5 = -7.5

Simplifying
n + -1.5 = -7.5

Reorder the terms:
-1.5 + n = -7.5

Solving
-1.5 + n = -7.5

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 = -7.5 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + n = -7.5 + 1.5
n = -7.5 + 1.5

Combine like terms: -7.5 + 1.5 = -6
n = -6

Simplifying
n = -6

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {9, -6}