500=.5(4x)(5x+1)

Solution for 500=.5(4x)(5x+1) equation:

Simplifying
500 = 0.5(4x)(5x + 1)

Remove parenthesis around (4x)
500 = 0.5 * 4x(5x + 1)

Reorder the terms:
500 = 0.5 * 4x(1 + 5x)

Multiply 0.5 * 4
500 = 2x(1 + 5x)
500 = (1 * 2x + 5x * 2x)
500 = (2x + 10x2)

Solving
500 = 2x + 10x2

Solving for variable 'x'.

Reorder the terms:
500 + -2x + -10x2 = 2x + -2x + 10x2 + -10x2

Combine like terms: 2x + -2x = 0
500 + -2x + -10x2 = 0 + 10x2 + -10x2
500 + -2x + -10x2 = 10x2 + -10x2

Combine like terms: 10x2 + -10x2 = 0
500 + -2x + -10x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(250 + -1x + -5x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(250 + -1x + -5x2)' equal to zero and attempt to solve:

Simplifying
250 + -1x + -5x2 = 0

Solving
250 + -1x + -5x2 = 0

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-50 + 0.2x + x2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + 0.2x + 50 + x2 = 0 + 50

Reorder the terms:
-50 + 50 + 0.2x + x2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + 0.2x + x2 = 0 + 50
0.2x + x2 = 0 + 50

Combine like terms: 0 + 50 = 50
0.2x + x2 = 50

The x term is 0.2x.  Take half its coefficient (0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
0.2x + 0.01 + x2 = 50 + 0.01

Reorder the terms:
0.01 + 0.2x + x2 = 50 + 0.01

Combine like terms: 50 + 0.01 = 50.01
0.01 + 0.2x + x2 = 50.01

Factor a perfect square on the left side:
(x + 0.1)(x + 0.1) = 50.01

Calculate the square root of the right side: 7.071774883

Break this problem into two subproblems by setting 
(x + 0.1) equal to 7.071774883 and -7.071774883.

Subproblem 1
x + 0.1 = 7.071774883

Simplifying
x + 0.1 = 7.071774883

Reorder the terms:
0.1 + x = 7.071774883

Solving
0.1 + x = 7.071774883

Solving for variable 'x'.

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

Add '-0.1' to each side of the equation.
0.1 + -0.1 + x = 7.071774883 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + x = 7.071774883 + -0.1
x = 7.071774883 + -0.1

Combine like terms: 7.071774883 + -0.1 = 6.971774883
x = 6.971774883

Simplifying
x = 6.971774883

Subproblem 2
x + 0.1 = -7.071774883

Simplifying
x + 0.1 = -7.071774883

Reorder the terms:
0.1 + x = -7.071774883

Solving
0.1 + x = -7.071774883

Solving for variable 'x'.

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

Add '-0.1' to each side of the equation.
0.1 + -0.1 + x = -7.071774883 + -0.1

Combine like terms: 0.1 + -0.1 = 0.0
0.0 + x = -7.071774883 + -0.1
x = -7.071774883 + -0.1

Combine like terms: -7.071774883 + -0.1 = -7.171774883
x = -7.171774883

Simplifying
x = -7.171774883

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.971774883, -7.171774883}
Solution
x = {6.971774883, -7.171774883}