100000=x(1000-2x)

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

Simplifying
100000 = x(1000 + -2x)
100000 = (1000 * x + -2x * x)
100000 = (1000x + -2x2)

Solving
100000 = 1000x + -2x2

Solving for variable 'x'.

Reorder the terms:
100000 + -1000x + 2x2 = 1000x + -1000x + -2x2 + 2x2

Combine like terms: 1000x + -1000x = 0
100000 + -1000x + 2x2 = 0 + -2x2 + 2x2
100000 + -1000x + 2x2 = -2x2 + 2x2

Combine like terms: -2x2 + 2x2 = 0
100000 + -1000x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(50000 + -500x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(50000 + -500x + x2)' equal to zero and attempt to solve:

Simplifying
50000 + -500x + x2 = 0

Solving
50000 + -500x + x2 = 0

Begin completing the square.

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}