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