X(1000-2X)=100000

Solution for X(1000-2X)=100000 equation:

Simplifying
X(1000 + -2X) = 100000
(1000 * X + -2X * X) = 100000
(1000X + -2X2) = 100000

Solving
1000X + -2X2 = 100000

Solving for variable 'X'.

Reorder the terms:
-100000 + 1000X + -2X2 = 100000 + -100000

Combine like terms: 100000 + -100000 = 0
-100000 + 1000X + -2X2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-50000 + 500X + -1X2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-50000 + 500X + -1X2)' equal to zero and attempt to solve:

Simplifying
-50000 + 500X + -1X2 = 0

Solving
-50000 + 500X + -1X2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
50000 + -500X + X2 = 0

Move the constant term to the right:

Add '-50000' to each side of the equation.
50000 + -500X + -50000 + X2 = 0 + -50000

Reorder the terms:
50000 + -50000 + -500X + X2 = 0 + -50000

Combine like terms: 50000 + -50000 = 0
0 + -500X + X2 = 0 + -50000
-500X + X2 = 0 + -50000

Combine like terms: 0 + -50000 = -50000
-500X + X2 = -50000

The X term is -500X.  Take half its coefficient (-250).
Square it (62500) and add it to both sides.

Add '62500' to each side of the equation.
-500X + 62500 + X2 = -50000 + 62500

Reorder the terms:
62500 + -500X + X2 = -50000 + 62500

Combine like terms: -50000 + 62500 = 12500
62500 + -500X + X2 = 12500

Factor a perfect square on the left side:
(X + -250)(X + -250) = 12500

Calculate the square root of the right side: 111.803398875

Break this problem into two subproblems by setting 
(X + -250) equal to 111.803398875 and -111.803398875.

Subproblem 1
X + -250 = 111.803398875

Simplifying
X + -250 = 111.803398875

Reorder the terms:
-250 + X = 111.803398875

Solving
-250 + X = 111.803398875

Solving for variable 'X'.

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

Add '250' to each side of the equation.
-250 + 250 + X = 111.803398875 + 250

Combine like terms: -250 + 250 = 0
0 + X = 111.803398875 + 250
X = 111.803398875 + 250

Combine like terms: 111.803398875 + 250 = 361.803398875
X = 361.803398875

Simplifying
X = 361.803398875

Subproblem 2
X + -250 = -111.803398875

Simplifying
X + -250 = -111.803398875

Reorder the terms:
-250 + X = -111.803398875

Solving
-250 + X = -111.803398875

Solving for variable 'X'.

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

Add '250' to each side of the equation.
-250 + 250 + X = -111.803398875 + 250

Combine like terms: -250 + 250 = 0
0 + X = -111.803398875 + 250
X = -111.803398875 + 250

Combine like terms: -111.803398875 + 250 = 138.196601125
X = 138.196601125

Simplifying
X = 138.196601125

Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {361.803398875, 138.196601125}
Solution
X = {361.803398875, 138.196601125}