x^2+1000x+0.25=0

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Solution for x^2+1000x+0.25=0 equation:


Simplifying
x2 + 1000x + 0.25 = 0

Reorder the terms:
0.25 + 1000x + x2 = 0

Solving
0.25 + 1000x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1000x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1000x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1000x + x2 = 0 + -0.25
1000x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1000x + x2 = -0.25

The x term is 1000x.  Take half its coefficient (500).
Square it (250000) and add it to both sides.

Add '250000' to each side of the equation.
1000x + 250000 + x2 = -0.25 + 250000

Reorder the terms:
250000 + 1000x + x2 = -0.25 + 250000

Combine like terms: -0.25 + 250000 = 249999.75
250000 + 1000x + x2 = 249999.75

Factor a perfect square on the left side:
(x + 500)(x + 500) = 249999.75

Calculate the square root of the right side: 499.99975

Break this problem into two subproblems by setting 
(x + 500) equal to 499.99975 and -499.99975.

Subproblem 1

x + 500 = 499.99975 Simplifying x + 500 = 499.99975 Reorder the terms: 500 + x = 499.99975 Solving 500 + x = 499.99975 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-500' to each side of the equation. 500 + -500 + x = 499.99975 + -500 Combine like terms: 500 + -500 = 0 0 + x = 499.99975 + -500 x = 499.99975 + -500 Combine like terms: 499.99975 + -500 = -0.00025 x = -0.00025 Simplifying x = -0.00025

Subproblem 2

x + 500 = -499.99975 Simplifying x + 500 = -499.99975 Reorder the terms: 500 + x = -499.99975 Solving 500 + x = -499.99975 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-500' to each side of the equation. 500 + -500 + x = -499.99975 + -500 Combine like terms: 500 + -500 = 0 0 + x = -499.99975 + -500 x = -499.99975 + -500 Combine like terms: -499.99975 + -500 = -999.99975 x = -999.99975 Simplifying x = -999.99975

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.00025, -999.99975}

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