1.5n^2-1.5n=3

Solution for 1.5n^2-1.5n=3 equation:

Simplifying
1.5n2 + -1.5n = 3

Reorder the terms:
-1.5n + 1.5n2 = 3

Solving
-1.5n + 1.5n2 = 3

Solving for variable 'n'.

Reorder the terms:
-3 + -1.5n + 1.5n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + -1.5n + 1.5n2 = 0

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

Divide each side by '1.5'.
-2 + -1n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -1n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1n + n2 = 0 + 2
-1n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1n + n2 = 2

The n term is -1n.  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.
-1n + 0.25 + n2 = 2 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 2 + 0.25

Combine like terms: 2 + 0.25 = 2.25
0.25 + -1n + n2 = 2.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 2.25

Calculate the square root of the right side: 1.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 1.5 and -1.5.

Subproblem 1
n + -0.5 = 1.5

Simplifying
n + -0.5 = 1.5

Reorder the terms:
-0.5 + n = 1.5

Solving
-0.5 + n = 1.5

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = 1.5 + 0.5

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

Combine like terms: 1.5 + 0.5 = 2
n = 2

Simplifying
n = 2

Subproblem 2
n + -0.5 = -1.5

Simplifying
n + -0.5 = -1.5

Reorder the terms:
-0.5 + n = -1.5

Solving
-0.5 + n = -1.5

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = -1.5 + 0.5

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

Combine like terms: -1.5 + 0.5 = -1
n = -1

Simplifying
n = -1

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2, -1}