5x+10x(90-x)=500

Solution for 5x+10x(90-x)=500 equation:

Simplifying
5x + 10x(90 + -1x) = 500
5x + (90 * 10x + -1x * 10x) = 500
5x + (900x + -10x2) = 500

Combine like terms: 5x + 900x = 905x
905x + -10x2 = 500

Solving
905x + -10x2 = 500

Solving for variable 'x'.

Reorder the terms:
-500 + 905x + -10x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + 905x + -10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-100 + 181x + -2x2) = 0

Ignore the factor 5.
Subproblem 1
Set the factor '(-100 + 181x + -2x2)' equal to zero and attempt to solve:

Simplifying
-100 + 181x + -2x2 = 0

Solving
-100 + 181x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
50 + -90.5x + x2 = 0

Move the constant term to the right:

Add '-50' to each side of the equation.
50 + -90.5x + -50 + x2 = 0 + -50

Reorder the terms:
50 + -50 + -90.5x + x2 = 0 + -50

Combine like terms: 50 + -50 = 0
0 + -90.5x + x2 = 0 + -50
-90.5x + x2 = 0 + -50

Combine like terms: 0 + -50 = -50
-90.5x + x2 = -50

The x term is -90.5x.  Take half its coefficient (-45.25).
Square it (2047.5625) and add it to both sides.

Add '2047.5625' to each side of the equation.
-90.5x + 2047.5625 + x2 = -50 + 2047.5625

Reorder the terms:
2047.5625 + -90.5x + x2 = -50 + 2047.5625

Combine like terms: -50 + 2047.5625 = 1997.5625
2047.5625 + -90.5x + x2 = 1997.5625

Factor a perfect square on the left side:
(x + -45.25)(x + -45.25) = 1997.5625

Calculate the square root of the right side: 44.694099163

Break this problem into two subproblems by setting 
(x + -45.25) equal to 44.694099163 and -44.694099163.

Subproblem 1
x + -45.25 = 44.694099163

Simplifying
x + -45.25 = 44.694099163

Reorder the terms:
-45.25 + x = 44.694099163

Solving
-45.25 + x = 44.694099163

Solving for variable 'x'.

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

Add '45.25' to each side of the equation.
-45.25 + 45.25 + x = 44.694099163 + 45.25

Combine like terms: -45.25 + 45.25 = 0.00
0.00 + x = 44.694099163 + 45.25
x = 44.694099163 + 45.25

Combine like terms: 44.694099163 + 45.25 = 89.944099163
x = 89.944099163

Simplifying
x = 89.944099163

Subproblem 2
x + -45.25 = -44.694099163

Simplifying
x + -45.25 = -44.694099163

Reorder the terms:
-45.25 + x = -44.694099163

Solving
-45.25 + x = -44.694099163

Solving for variable 'x'.

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

Add '45.25' to each side of the equation.
-45.25 + 45.25 + x = -44.694099163 + 45.25

Combine like terms: -45.25 + 45.25 = 0.00
0.00 + x = -44.694099163 + 45.25
x = -44.694099163 + 45.25

Combine like terms: -44.694099163 + 45.25 = 0.555900837
x = 0.555900837

Simplifying
x = 0.555900837

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {89.944099163, 0.555900837}
Solution
x = {89.944099163, 0.555900837}