0=-10000+5000x-500x^2

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Solution for 0=-10000+5000x-500x^2 equation: 


Simplifying
0 = -10000 + 5000x + -500x2

Solving
0 = -10000 + 5000x + -500x2

Solving for variable 'x'.

Combine like terms: 0 + 10000 = 10000
10000 + -5000x + 500x2 = -10000 + 5000x + -500x2 + 10000 + -5000x + 500x2

Reorder the terms:
10000 + -5000x + 500x2 = -10000 + 10000 + 5000x + -5000x + -500x2 + 500x2

Combine like terms: -10000 + 10000 = 0
10000 + -5000x + 500x2 = 0 + 5000x + -5000x + -500x2 + 500x2
10000 + -5000x + 500x2 = 5000x + -5000x + -500x2 + 500x2

Combine like terms: 5000x + -5000x = 0
10000 + -5000x + 500x2 = 0 + -500x2 + 500x2
10000 + -5000x + 500x2 = -500x2 + 500x2

Combine like terms: -500x2 + 500x2 = 0
10000 + -5000x + 500x2 = 0

Factor out the Greatest Common Factor (GCF), '500'.
500(20 + -10x + x2) = 0

Ignore the factor 500.

Subproblem 1
Set the factor '(20 + -10x + x2)' equal to zero and attempt to solve:

Simplifying
20 + -10x + x2 = 0

Solving
20 + -10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + -10x + -20 + x2 = 0 + -20

Reorder the terms:
20 + -20 + -10x + x2 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + -10x + x2 = 0 + -20
-10x + x2 = 0 + -20

Combine like terms: 0 + -20 = -20
-10x + x2 = -20

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -20 + 25

Reorder the terms:
25 + -10x + x2 = -20 + 25

Combine like terms: -20 + 25 = 5
25 + -10x + x2 = 5

Factor a perfect square on the left side:
(x + -5)(x + -5) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + -5) equal to 2.236067978 and -2.236067978.


Subproblem 1
x + -5 = 2.236067978

Simplifying
x + -5 = 2.236067978

Reorder the terms:
-5 + x = 2.236067978

Solving
-5 + x = 2.236067978

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = 2.236067978 + 5

Combine like terms: -5 + 5 = 0
0 + x = 2.236067978 + 5
x = 2.236067978 + 5

Combine like terms: 2.236067978 + 5 = 7.236067978
x = 7.236067978

Simplifying
x = 7.236067978


Subproblem 2
x + -5 = -2.236067978

Simplifying
x + -5 = -2.236067978

Reorder the terms:
-5 + x = -2.236067978

Solving
-5 + x = -2.236067978

Solving for variable 'x'.

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

Add '5' to each side of the equation.
-5 + 5 + x = -2.236067978 + 5

Combine like terms: -5 + 5 = 0
0 + x = -2.236067978 + 5
x = -2.236067978 + 5

Combine like terms: -2.236067978 + 5 = 2.763932022
x = 2.763932022

Simplifying
x = 2.763932022


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.236067978, 2.763932022}
Solution
x = {7.236067978, 2.763932022}

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