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

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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 = -895x
-895x + 10x2 = 500

Solving
-895x + 10x2 = 500

Solving for variable 'x'.

Reorder the terms:
-500 + -895x + 10x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + -895x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-100 + -179x + 2x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-100 + -179x + 2x2)' equal to zero and attempt to solve:

Simplifying
-100 + -179x + 2x2 = 0

Solving
-100 + -179x + 2x2 = 0

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

Divide each side by '2'.
-50 + -89.5x + x2 = 0

Move the constant term to the right:

Add '50' to each side of the equation.
-50 + -89.5x + 50 + x2 = 0 + 50

Reorder the terms:
-50 + 50 + -89.5x + x2 = 0 + 50

Combine like terms: -50 + 50 = 0
0 + -89.5x + x2 = 0 + 50
-89.5x + x2 = 0 + 50

Combine like terms: 0 + 50 = 50
-89.5x + x2 = 50

The x term is -89.5x.  Take half its coefficient (-44.75).
Square it (2002.5625) and add it to both sides.

Add '2002.5625' to each side of the equation.
-89.5x + 2002.5625 + x2 = 50 + 2002.5625

Reorder the terms:
2002.5625 + -89.5x + x2 = 50 + 2002.5625

Combine like terms: 50 + 2002.5625 = 2052.5625
2002.5625 + -89.5x + x2 = 2052.5625

Factor a perfect square on the left side:
(x + -44.75)(x + -44.75) = 2052.5625

Calculate the square root of the right side: 45.305214932

Break this problem into two subproblems by setting 
(x + -44.75) equal to 45.305214932 and -45.305214932.


Subproblem 1
x + -44.75 = 45.305214932

Simplifying
x + -44.75 = 45.305214932

Reorder the terms:
-44.75 + x = 45.305214932

Solving
-44.75 + x = 45.305214932

Solving for variable 'x'.

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

Add '44.75' to each side of the equation.
-44.75 + 44.75 + x = 45.305214932 + 44.75

Combine like terms: -44.75 + 44.75 = 0.00
0.00 + x = 45.305214932 + 44.75
x = 45.305214932 + 44.75

Combine like terms: 45.305214932 + 44.75 = 90.055214932
x = 90.055214932

Simplifying
x = 90.055214932


Subproblem 2
x + -44.75 = -45.305214932

Simplifying
x + -44.75 = -45.305214932

Reorder the terms:
-44.75 + x = -45.305214932

Solving
-44.75 + x = -45.305214932

Solving for variable 'x'.

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

Add '44.75' to each side of the equation.
-44.75 + 44.75 + x = -45.305214932 + 44.75

Combine like terms: -44.75 + 44.75 = 0.00
0.00 + x = -45.305214932 + 44.75
x = -45.305214932 + 44.75

Combine like terms: -45.305214932 + 44.75 = -0.555214932
x = -0.555214932

Simplifying
x = -0.555214932


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {90.055214932, -0.555214932}
Solution
x = {90.055214932, -0.555214932}

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