x-0.001x^2=-200

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Solution for x-0.001x^2=-200 equation:


Simplifying
x + -0.001x2 = -200

Solving
x + -0.001x2 = -200

Solving for variable 'x'.

Reorder the terms:
200 + x + -0.001x2 = -200 + 200

Combine like terms: -200 + 200 = 0
200 + x + -0.001x2 = 0

Begin completing the square.  Divide all terms by
-0.001 the coefficient of the squared term: 

Divide each side by '-0.001'.
-200000 + -1000x + x2 = 0

Move the constant term to the right:

Add '200000' to each side of the equation.
-200000 + -1000x + 200000 + x2 = 0 + 200000

Reorder the terms:
-200000 + 200000 + -1000x + x2 = 0 + 200000

Combine like terms: -200000 + 200000 = 0
0 + -1000x + x2 = 0 + 200000
-1000x + x2 = 0 + 200000

Combine like terms: 0 + 200000 = 200000
-1000x + x2 = 200000

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1000x + 0.25 + x2 = 200000 + 0.25

Reorder the terms:
0.25 + -1000x + x2 = 200000 + 0.25

Combine like terms: 200000 + 0.25 = 200000.25
0.25 + -1000x + x2 = 200000.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 200000.25

Calculate the square root of the right side: 447.213875008

Break this problem into two subproblems by setting 
(x + 0.5) equal to 447.213875008 and -447.213875008.

Subproblem 1

x + 0.5 = 447.213875008 Simplifying x + 0.5 = 447.213875008 Reorder the terms: 0.5 + x = 447.213875008 Solving 0.5 + x = 447.213875008 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 447.213875008 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 447.213875008 + -0.5 x = 447.213875008 + -0.5 Combine like terms: 447.213875008 + -0.5 = 446.713875008 x = 446.713875008 Simplifying x = 446.713875008

Subproblem 2

x + 0.5 = -447.213875008 Simplifying x + 0.5 = -447.213875008 Reorder the terms: 0.5 + x = -447.213875008 Solving 0.5 + x = -447.213875008 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -447.213875008 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -447.213875008 + -0.5 x = -447.213875008 + -0.5 Combine like terms: -447.213875008 + -0.5 = -447.713875008 x = -447.713875008 Simplifying x = -447.713875008

Solution

The solution to the problem is based on the solutions from the subproblems. x = {446.713875008, -447.713875008}

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