15.75(x+0.5)(x+1.5)=48(2x)(x+5)

Solution for 15.75(x+0.5)(x+1.5)=48(2x)(x+5) equation:

Simplifying
15.75(x + 0.5)(x + 1.5) = 48(2x)(x + 5)

Reorder the terms:
15.75(0.5 + x)(x + 1.5) = 48(2x)(x + 5)

Reorder the terms:
15.75(0.5 + x)(1.5 + x) = 48(2x)(x + 5)

Multiply (0.5 + x) * (1.5 + x)
15.75(0.5(1.5 + x) + x(1.5 + x)) = 48(2x)(x + 5)
15.75((1.5 * 0.5 + x * 0.5) + x(1.5 + x)) = 48(2x)(x + 5)
15.75((0.75 + 0.5x) + x(1.5 + x)) = 48(2x)(x + 5)
15.75(0.75 + 0.5x + (1.5 * x + x * x)) = 48(2x)(x + 5)
15.75(0.75 + 0.5x + (1.5x + x2)) = 48(2x)(x + 5)

Combine like terms: 0.5x + 1.5x = 2x
15.75(0.75 + 2x + x2) = 48(2x)(x + 5)
(0.75 * 15.75 + 2x * 15.75 + x2 * 15.75) = 48(2x)(x + 5)
(11.8125 + 31.5x + 15.75x2) = 48(2x)(x + 5)

Remove parenthesis around (2x)
11.8125 + 31.5x + 15.75x2 = 48 * 2x(x + 5)

Reorder the terms:
11.8125 + 31.5x + 15.75x2 = 48 * 2x(5 + x)

Multiply 48 * 2
11.8125 + 31.5x + 15.75x2 = 96x(5 + x)
11.8125 + 31.5x + 15.75x2 = (5 * 96x + x * 96x)
11.8125 + 31.5x + 15.75x2 = (480x + 96x2)

Solving
11.8125 + 31.5x + 15.75x2 = 480x + 96x2

Solving for variable 'x'.

Reorder the terms:
11.8125 + 31.5x + -480x + 15.75x2 + -96x2 = 480x + 96x2 + -480x + -96x2

Combine like terms: 31.5x + -480x = -448.5x
11.8125 + -448.5x + 15.75x2 + -96x2 = 480x + 96x2 + -480x + -96x2

Combine like terms: 15.75x2 + -96x2 = -80.25x2
11.8125 + -448.5x + -80.25x2 = 480x + 96x2 + -480x + -96x2

Reorder the terms:
11.8125 + -448.5x + -80.25x2 = 480x + -480x + 96x2 + -96x2

Combine like terms: 480x + -480x = 0
11.8125 + -448.5x + -80.25x2 = 0 + 96x2 + -96x2
11.8125 + -448.5x + -80.25x2 = 96x2 + -96x2

Combine like terms: 96x2 + -96x2 = 0
11.8125 + -448.5x + -80.25x2 = 0

Begin completing the square.  Divide all terms by
-80.25 the coefficient of the squared term: 

Divide each side by '-80.25'.
-0.1471962617 + 5.588785047x + x2 = 0

Move the constant term to the right:

Add '0.1471962617' to each side of the equation.
-0.1471962617 + 5.588785047x + 0.1471962617 + x2 = 0 + 0.1471962617

Reorder the terms:
-0.1471962617 + 0.1471962617 + 5.588785047x + x2 = 0 + 0.1471962617

Combine like terms: -0.1471962617 + 0.1471962617 = 0.0000000000
0.0000000000 + 5.588785047x + x2 = 0 + 0.1471962617
5.588785047x + x2 = 0 + 0.1471962617

Combine like terms: 0 + 0.1471962617 = 0.1471962617
5.588785047x + x2 = 0.1471962617

The x term is 5.588785047x.  Take half its coefficient (2.794392524).
Square it (7.808629578) and add it to both sides.

Add '7.808629578' to each side of the equation.
5.588785047x + 7.808629578 + x2 = 0.1471962617 + 7.808629578

Reorder the terms:
7.808629578 + 5.588785047x + x2 = 0.1471962617 + 7.808629578

Combine like terms: 0.1471962617 + 7.808629578 = 7.9558258397
7.808629578 + 5.588785047x + x2 = 7.9558258397

Factor a perfect square on the left side:
(x + 2.794392524)(x + 2.794392524) = 7.9558258397

Calculate the square root of the right side: 2.820607353

Break this problem into two subproblems by setting 
(x + 2.794392524) equal to 2.820607353 and -2.820607353.

Subproblem 1
x + 2.794392524 = 2.820607353

Simplifying
x + 2.794392524 = 2.820607353

Reorder the terms:
2.794392524 + x = 2.820607353

Solving
2.794392524 + x = 2.820607353

Solving for variable 'x'.

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

Add '-2.794392524' to each side of the equation.
2.794392524 + -2.794392524 + x = 2.820607353 + -2.794392524

Combine like terms: 2.794392524 + -2.794392524 = 0.000000000
0.000000000 + x = 2.820607353 + -2.794392524
x = 2.820607353 + -2.794392524

Combine like terms: 2.820607353 + -2.794392524 = 0.026214829
x = 0.026214829

Simplifying
x = 0.026214829

Subproblem 2
x + 2.794392524 = -2.820607353

Simplifying
x + 2.794392524 = -2.820607353

Reorder the terms:
2.794392524 + x = -2.820607353

Solving
2.794392524 + x = -2.820607353

Solving for variable 'x'.

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

Add '-2.794392524' to each side of the equation.
2.794392524 + -2.794392524 + x = -2.820607353 + -2.794392524

Combine like terms: 2.794392524 + -2.794392524 = 0.000000000
0.000000000 + x = -2.820607353 + -2.794392524
x = -2.820607353 + -2.794392524

Combine like terms: -2.820607353 + -2.794392524 = -5.614999877
x = -5.614999877

Simplifying
x = -5.614999877

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.026214829, -5.614999877}