=(24x-1.5x^2)-(.5x^2+4x+8)

Solution for =(24x-1.5x^2)-(.5x^2+4x+8) equation:

Simplifying
0 = (24x + -1.5x2) + -1(0.5x2 + 4x + 8)

Remove parenthesis around (24x + -1.5x2)
0 = 24x + -1.5x2 + -1(0.5x2 + 4x + 8)

Reorder the terms:
0 = 24x + -1.5x2 + -1(8 + 4x + 0.5x2)
0 = 24x + -1.5x2 + (8 * -1 + 4x * -1 + 0.5x2 * -1)
0 = 24x + -1.5x2 + (-8 + -4x + -0.5x2)

Reorder the terms:
0 = -8 + 24x + -4x + -1.5x2 + -0.5x2

Combine like terms: 24x + -4x = 20x
0 = -8 + 20x + -1.5x2 + -0.5x2

Combine like terms: -1.5x2 + -0.5x2 = -2x2
0 = -8 + 20x + -2x2

Solving
0 = -8 + 20x + -2x2

Solving for variable 'x'.

Combine like terms: 0 + 8 = 8
8 + -20x + 2x2 = -8 + 20x + -2x2 + 8 + -20x + 2x2

Reorder the terms:
8 + -20x + 2x2 = -8 + 8 + 20x + -20x + -2x2 + 2x2

Combine like terms: -8 + 8 = 0
8 + -20x + 2x2 = 0 + 20x + -20x + -2x2 + 2x2
8 + -20x + 2x2 = 20x + -20x + -2x2 + 2x2

Combine like terms: 20x + -20x = 0
8 + -20x + 2x2 = 0 + -2x2 + 2x2
8 + -20x + 2x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
8 + -20x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(4 + -10x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(4 + -10x + x2)' equal to zero and attempt to solve:

Simplifying
4 + -10x + x2 = 0

Solving
4 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -10x + -4 + x2 = 0 + -4

Reorder the terms:
4 + -4 + -10x + x2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -10x + x2 = 0 + -4
-10x + x2 = 0 + -4

Combine like terms: 0 + -4 = -4
-10x + x2 = -4

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -4 + 25

Reorder the terms:
25 + -10x + x2 = -4 + 25

Combine like terms: -4 + 25 = 21
25 + -10x + x2 = 21

Factor a perfect square on the left side:
(x + -5)(x + -5) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(x + -5) equal to 4.582575695 and -4.582575695.

Subproblem 1
x + -5 = 4.582575695

Simplifying
x + -5 = 4.582575695

Reorder the terms:
-5 + x = 4.582575695

Solving
-5 + x = 4.582575695

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = 4.582575695 + 5

Combine like terms: -5 + 5 = 0
0 + x = 4.582575695 + 5
x = 4.582575695 + 5

Combine like terms: 4.582575695 + 5 = 9.582575695
x = 9.582575695

Simplifying
x = 9.582575695

Subproblem 2
x + -5 = -4.582575695

Simplifying
x + -5 = -4.582575695

Reorder the terms:
-5 + x = -4.582575695

Solving
-5 + x = -4.582575695

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = -4.582575695 + 5

Combine like terms: -5 + 5 = 0
0 + x = -4.582575695 + 5
x = -4.582575695 + 5

Combine like terms: -4.582575695 + 5 = 0.417424305
x = 0.417424305

Simplifying
x = 0.417424305

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.582575695, 0.417424305}
Solution
x = {9.582575695, 0.417424305}