24(x+0.5)(x+1.5)=7.875(2x)(x+5)

Solution for 24(x+0.5)(x+1.5)=7.875(2x)(x+5) equation:

Simplifying
24(x + 0.5)(x + 1.5) = 7.875(2x)(x + 5)

Reorder the terms:
24(0.5 + x)(x + 1.5) = 7.875(2x)(x + 5)

Reorder the terms:
24(0.5 + x)(1.5 + x) = 7.875(2x)(x + 5)

Multiply (0.5 + x) * (1.5 + x)
24(0.5(1.5 + x) + x(1.5 + x)) = 7.875(2x)(x + 5)
24((1.5 * 0.5 + x * 0.5) + x(1.5 + x)) = 7.875(2x)(x + 5)
24((0.75 + 0.5x) + x(1.5 + x)) = 7.875(2x)(x + 5)
24(0.75 + 0.5x + (1.5 * x + x * x)) = 7.875(2x)(x + 5)
24(0.75 + 0.5x + (1.5x + x2)) = 7.875(2x)(x + 5)

Combine like terms: 0.5x + 1.5x = 2x
24(0.75 + 2x + x2) = 7.875(2x)(x + 5)
(0.75 * 24 + 2x * 24 + x2 * 24) = 7.875(2x)(x + 5)
(18 + 48x + 24x2) = 7.875(2x)(x + 5)

Remove parenthesis around (2x)
18 + 48x + 24x2 = 7.875 * 2x(x + 5)

Reorder the terms:
18 + 48x + 24x2 = 7.875 * 2x(5 + x)

Multiply 7.875 * 2
18 + 48x + 24x2 = 15.75x(5 + x)
18 + 48x + 24x2 = (5 * 15.75x + x * 15.75x)
18 + 48x + 24x2 = (78.75x + 15.75x2)

Solving
18 + 48x + 24x2 = 78.75x + 15.75x2

Solving for variable 'x'.

Reorder the terms:
18 + 48x + -78.75x + 24x2 + -15.75x2 = 78.75x + 15.75x2 + -78.75x + -15.75x2

Combine like terms: 48x + -78.75x = -30.75x
18 + -30.75x + 24x2 + -15.75x2 = 78.75x + 15.75x2 + -78.75x + -15.75x2

Combine like terms: 24x2 + -15.75x2 = 8.25x2
18 + -30.75x + 8.25x2 = 78.75x + 15.75x2 + -78.75x + -15.75x2

Reorder the terms:
18 + -30.75x + 8.25x2 = 78.75x + -78.75x + 15.75x2 + -15.75x2

Combine like terms: 78.75x + -78.75x = 0.00
18 + -30.75x + 8.25x2 = 0.00 + 15.75x2 + -15.75x2
18 + -30.75x + 8.25x2 = 15.75x2 + -15.75x2

Combine like terms: 15.75x2 + -15.75x2 = 0.00
18 + -30.75x + 8.25x2 = 0.00

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

Divide each side by '8.25'.
2.181818182 + -3.727272727x + x2 = 0

Move the constant term to the right:

Add '-2.181818182' to each side of the equation.
2.181818182 + -3.727272727x + -2.181818182 + x2 = 0 + -2.181818182

Reorder the terms:
2.181818182 + -2.181818182 + -3.727272727x + x2 = 0 + -2.181818182

Combine like terms: 2.181818182 + -2.181818182 = 0.000000000
0.000000000 + -3.727272727x + x2 = 0 + -2.181818182
-3.727272727x + x2 = 0 + -2.181818182

Combine like terms: 0 + -2.181818182 = -2.181818182
-3.727272727x + x2 = -2.181818182

The x term is -3.727272727x.  Take half its coefficient (-1.863636364).
Square it (3.473140497) and add it to both sides.

Add '3.473140497' to each side of the equation.
-3.727272727x + 3.473140497 + x2 = -2.181818182 + 3.473140497

Reorder the terms:
3.473140497 + -3.727272727x + x2 = -2.181818182 + 3.473140497

Combine like terms: -2.181818182 + 3.473140497 = 1.291322315
3.473140497 + -3.727272727x + x2 = 1.291322315

Factor a perfect square on the left side:
(x + -1.863636364)(x + -1.863636364) = 1.291322315

Calculate the square root of the right side: 1.136363637

Break this problem into two subproblems by setting 
(x + -1.863636364) equal to 1.136363637 and -1.136363637.

Subproblem 1
x + -1.863636364 = 1.136363637

Simplifying
x + -1.863636364 = 1.136363637

Reorder the terms:
-1.863636364 + x = 1.136363637

Solving
-1.863636364 + x = 1.136363637

Solving for variable 'x'.

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

Add '1.863636364' to each side of the equation.
-1.863636364 + 1.863636364 + x = 1.136363637 + 1.863636364

Combine like terms: -1.863636364 + 1.863636364 = 0.000000000
0.000000000 + x = 1.136363637 + 1.863636364
x = 1.136363637 + 1.863636364

Combine like terms: 1.136363637 + 1.863636364 = 3.000000001
x = 3.000000001

Simplifying
x = 3.000000001

Subproblem 2
x + -1.863636364 = -1.136363637

Simplifying
x + -1.863636364 = -1.136363637

Reorder the terms:
-1.863636364 + x = -1.136363637

Solving
-1.863636364 + x = -1.136363637

Solving for variable 'x'.

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

Add '1.863636364' to each side of the equation.
-1.863636364 + 1.863636364 + x = -1.136363637 + 1.863636364

Combine like terms: -1.863636364 + 1.863636364 = 0.000000000
0.000000000 + x = -1.136363637 + 1.863636364
x = -1.136363637 + 1.863636364

Combine like terms: -1.136363637 + 1.863636364 = 0.727272727
x = 0.727272727

Simplifying
x = 0.727272727

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.000000001, 0.727272727}