1.5x^2-4.5x+3=0

Solution for 1.5x^2-4.5x+3=0 equation:

Simplifying
1.5x2 + -4.5x + 3 = 0

Reorder the terms:
3 + -4.5x + 1.5x2 = 0

Solving
3 + -4.5x + 1.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.5'.
2 + -3x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -3x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + -3x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -3x + x2 = 0 + -2
-3x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
-3x + x2 = -2

The x term is -3x.  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.
-3x + 2.25 + x2 = -2 + 2.25

Reorder the terms:
2.25 + -3x + x2 = -2 + 2.25

Combine like terms: -2 + 2.25 = 0.25
2.25 + -3x + x2 = 0.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 0.25

Calculate the square root of the right side: 0.5

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

Subproblem 1
x + -1.5 = 0.5

Simplifying
x + -1.5 = 0.5

Reorder the terms:
-1.5 + x = 0.5

Solving
-1.5 + x = 0.5

Solving for variable 'x'.

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

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

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

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

Simplifying
x = 2

Subproblem 2
x + -1.5 = -0.5

Simplifying
x + -1.5 = -0.5

Reorder the terms:
-1.5 + x = -0.5

Solving
-1.5 + x = -0.5

Solving for variable 'x'.

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

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

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

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

Simplifying
x = 1

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