0.001x^2+1000x+0.001=0

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Solution for 0.001x^2+1000x+0.001=0 equation: 


Simplifying
0.001x2 + 1000x + 0.001 = 0

Reorder the terms:
0.001 + 1000x + 0.001x2 = 0

Solving
0.001 + 1000x + 0.001x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.001'.
1 + 1000000x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 1000000x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 1000000x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 1000000x + x2 = 0 + -1
1000000x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
1000000x + x2 = -1

The x term is 1000000x.  Take half its coefficient (500000).
Square it (250000000000) and add it to both sides.

Add '250000000000' to each side of the equation.
1000000x + 250000000000 + x2 = -1 + 250000000000

Reorder the terms:
250000000000 + 1000000x + x2 = -1 + 250000000000

Combine like terms: -1 + 250000000000 = 249999999999
250000000000 + 1000000x + x2 = 249999999999

Factor a perfect square on the left side:
(x + 500000)(x + 500000) = 249999999999

Calculate the square root of the right side: 499999.999999

Break this problem into two subproblems by setting 
(x + 500000) equal to 499999.999999 and -499999.999999.


Subproblem 1
x + 500000 = 499999.999999

Simplifying
x + 500000 = 499999.999999

Reorder the terms:
500000 + x = 499999.999999

Solving
500000 + x = 499999.999999

Solving for variable 'x'.

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

Add '-500000' to each side of the equation.
500000 + -500000 + x = 499999.999999 + -500000

Combine like terms: 500000 + -500000 = 0
0 + x = 499999.999999 + -500000
x = 499999.999999 + -500000

Combine like terms: 499999.999999 + -500000 = -0.000001
x = -0.000001

Simplifying
x = -0.000001


Subproblem 2
x + 500000 = -499999.999999

Simplifying
x + 500000 = -499999.999999

Reorder the terms:
500000 + x = -499999.999999

Solving
500000 + x = -499999.999999

Solving for variable 'x'.

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

Add '-500000' to each side of the equation.
500000 + -500000 + x = -499999.999999 + -500000

Combine like terms: 500000 + -500000 = 0
0 + x = -499999.999999 + -500000
x = -499999.999999 + -500000

Combine like terms: -499999.999999 + -500000 = -999999.999999
x = -999999.999999

Simplifying
x = -999999.999999


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.000001, -999999.999999}

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