100x-x^2=500

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Solution for 100x-x^2=500 equation:


Simplifying
100x + -1x2 = 500

Solving
100x + -1x2 = 500

Solving for variable 'x'.

Reorder the terms:
-500 + 100x + -1x2 = 500 + -500

Combine like terms: 500 + -500 = 0
-500 + 100x + -1x2 = 0

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

Divide each side by '-1'.
500 + -100x + x2 = 0

Move the constant term to the right:

Add '-500' to each side of the equation.
500 + -100x + -500 + x2 = 0 + -500

Reorder the terms:
500 + -500 + -100x + x2 = 0 + -500

Combine like terms: 500 + -500 = 0
0 + -100x + x2 = 0 + -500
-100x + x2 = 0 + -500

Combine like terms: 0 + -500 = -500
-100x + x2 = -500

The x term is -100x.  Take half its coefficient (-50).
Square it (2500) and add it to both sides.

Add '2500' to each side of the equation.
-100x + 2500 + x2 = -500 + 2500

Reorder the terms:
2500 + -100x + x2 = -500 + 2500

Combine like terms: -500 + 2500 = 2000
2500 + -100x + x2 = 2000

Factor a perfect square on the left side:
(x + -50)(x + -50) = 2000

Calculate the square root of the right side: 44.72135955

Break this problem into two subproblems by setting 
(x + -50) equal to 44.72135955 and -44.72135955.

Subproblem 1

x + -50 = 44.72135955 Simplifying x + -50 = 44.72135955 Reorder the terms: -50 + x = 44.72135955 Solving -50 + x = 44.72135955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '50' to each side of the equation. -50 + 50 + x = 44.72135955 + 50 Combine like terms: -50 + 50 = 0 0 + x = 44.72135955 + 50 x = 44.72135955 + 50 Combine like terms: 44.72135955 + 50 = 94.72135955 x = 94.72135955 Simplifying x = 94.72135955

Subproblem 2

x + -50 = -44.72135955 Simplifying x + -50 = -44.72135955 Reorder the terms: -50 + x = -44.72135955 Solving -50 + x = -44.72135955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '50' to each side of the equation. -50 + 50 + x = -44.72135955 + 50 Combine like terms: -50 + 50 = 0 0 + x = -44.72135955 + 50 x = -44.72135955 + 50 Combine like terms: -44.72135955 + 50 = 5.27864045 x = 5.27864045 Simplifying x = 5.27864045

Solution

The solution to the problem is based on the solutions from the subproblems. x = {94.72135955, 5.27864045}

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