x(2x+54)=x-x+500

Solution for x(2x+54)=x-x+500 equation:

Simplifying
x(2x + 54) = x + -1x + 500

Reorder the terms:
x(54 + 2x) = x + -1x + 500
(54 * x + 2x * x) = x + -1x + 500
(54x + 2x2) = x + -1x + 500

Reorder the terms:
54x + 2x2 = 500 + x + -1x

Combine like terms: x + -1x = 0
54x + 2x2 = 500 + 0
54x + 2x2 = 500

Solving
54x + 2x2 = 500

Solving for variable 'x'.

Reorder the terms:
-500 + 54x + 2x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + 54x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-250 + 27x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-250 + 27x + x2)' equal to zero and attempt to solve:

Simplifying
-250 + 27x + x2 = 0

Solving
-250 + 27x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '250' to each side of the equation.
-250 + 27x + 250 + x2 = 0 + 250

Reorder the terms:
-250 + 250 + 27x + x2 = 0 + 250

Combine like terms: -250 + 250 = 0
0 + 27x + x2 = 0 + 250
27x + x2 = 0 + 250

Combine like terms: 0 + 250 = 250
27x + x2 = 250

The x term is 27x.  Take half its coefficient (13.5).
Square it (182.25) and add it to both sides.

Add '182.25' to each side of the equation.
27x + 182.25 + x2 = 250 + 182.25

Reorder the terms:
182.25 + 27x + x2 = 250 + 182.25

Combine like terms: 250 + 182.25 = 432.25
182.25 + 27x + x2 = 432.25

Factor a perfect square on the left side:
(x + 13.5)(x + 13.5) = 432.25

Calculate the square root of the right side: 20.790622886

Break this problem into two subproblems by setting 
(x + 13.5) equal to 20.790622886 and -20.790622886.

Subproblem 1
x + 13.5 = 20.790622886

Simplifying
x + 13.5 = 20.790622886

Reorder the terms:
13.5 + x = 20.790622886

Solving
13.5 + x = 20.790622886

Solving for variable 'x'.

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

Add '-13.5' to each side of the equation.
13.5 + -13.5 + x = 20.790622886 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + x = 20.790622886 + -13.5
x = 20.790622886 + -13.5

Combine like terms: 20.790622886 + -13.5 = 7.290622886
x = 7.290622886

Simplifying
x = 7.290622886

Subproblem 2
x + 13.5 = -20.790622886

Simplifying
x + 13.5 = -20.790622886

Reorder the terms:
13.5 + x = -20.790622886

Solving
13.5 + x = -20.790622886

Solving for variable 'x'.

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

Add '-13.5' to each side of the equation.
13.5 + -13.5 + x = -20.790622886 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + x = -20.790622886 + -13.5
x = -20.790622886 + -13.5

Combine like terms: -20.790622886 + -13.5 = -34.290622886
x = -34.290622886

Simplifying
x = -34.290622886

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.290622886, -34.290622886}
Solution
x = {7.290622886, -34.290622886}