0.5x(3-4x)=2-1.5(6+3x)

Solution for 0.5x(3-4x)=2-1.5(6+3x) equation:

Simplifying
0.5x(3 + -4x) = 2 + -1.5(6 + 3x)
(3 * 0.5x + -4x * 0.5x) = 2 + -1.5(6 + 3x)
(1.5x + -2x2) = 2 + -1.5(6 + 3x)
1.5x + -2x2 = 2 + (6 * -1.5 + 3x * -1.5)
1.5x + -2x2 = 2 + (-9 + -4.5x)

Combine like terms: 2 + -9 = -7
1.5x + -2x2 = -7 + -4.5x

Solving
1.5x + -2x2 = -7 + -4.5x

Solving for variable 'x'.

Reorder the terms:
7 + 1.5x + 4.5x + -2x2 = -7 + -4.5x + 7 + 4.5x

Combine like terms: 1.5x + 4.5x = 6x
7 + 6x + -2x2 = -7 + -4.5x + 7 + 4.5x

Reorder the terms:
7 + 6x + -2x2 = -7 + 7 + -4.5x + 4.5x

Combine like terms: -7 + 7 = 0
7 + 6x + -2x2 = 0 + -4.5x + 4.5x
7 + 6x + -2x2 = -4.5x + 4.5x

Combine like terms: -4.5x + 4.5x = 0.0
7 + 6x + -2x2 = 0.0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-3.5 + -3x + x2 = 0.0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + -3x + 3.5 + x2 = 0.0 + 3.5

Reorder the terms:
-3.5 + 3.5 + -3x + x2 = 0.0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + -3x + x2 = 0.0 + 3.5
-3x + x2 = 0.0 + 3.5

Combine like terms: 0.0 + 3.5 = 3.5
-3x + x2 = 3.5

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 = 3.5 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 3.5 + 2.25

Combine like terms: 3.5 + 2.25 = 5.75
2.25 + -3x + x2 = 5.75

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 5.75

Calculate the square root of the right side: 2.397915762

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.397915762 and -2.397915762.

Subproblem 1
x + -1.5 = 2.397915762

Simplifying
x + -1.5 = 2.397915762

Reorder the terms:
-1.5 + x = 2.397915762

Solving
-1.5 + x = 2.397915762

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 = 2.397915762 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 2.397915762 + 1.5
x = 2.397915762 + 1.5

Combine like terms: 2.397915762 + 1.5 = 3.897915762
x = 3.897915762

Simplifying
x = 3.897915762

Subproblem 2
x + -1.5 = -2.397915762

Simplifying
x + -1.5 = -2.397915762

Reorder the terms:
-1.5 + x = -2.397915762

Solving
-1.5 + x = -2.397915762

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 = -2.397915762 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -2.397915762 + 1.5
x = -2.397915762 + 1.5

Combine like terms: -2.397915762 + 1.5 = -0.897915762
x = -0.897915762

Simplifying
x = -0.897915762

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.897915762, -0.897915762}