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Simplifying x2 + 0.000413x + -0.000541 = 0 Reorder the terms: -0.000541 + 0.000413x + x2 = 0 Solving -0.000541 + 0.000413x + x2 = 0 Solving for variable 'x'. Begin completing the square. Move the constant term to the right: Add '0.000541' to each side of the equation. -0.000541 + 0.000413x + 0.000541 + x2 = 0 + 0.000541 Reorder the terms: -0.000541 + 0.000541 + 0.000413x + x2 = 0 + 0.000541 Combine like terms: -0.000541 + 0.000541 = 0.000000 0.000000 + 0.000413x + x2 = 0 + 0.000541 0.000413x + x2 = 0 + 0.000541 Combine like terms: 0 + 0.000541 = 0.000541 0.000413x + x2 = 0.000541 The x term is 0.000413x. Take half its coefficient (0.0002065). Square it (0.00000004264225) and add it to both sides. Add '0.00000004264225' to each side of the equation. 0.000413x + 0.00000004264225 + x2 = 0.000541 + 0.00000004264225 Reorder the terms: 0.00000004264225 + 0.000413x + x2 = 0.000541 + 0.00000004264225 Combine like terms: 0.000541 + 0.00000004264225 = 0.00054104264225 0.00000004264225 + 0.000413x + x2 = 0.00054104264225 Factor a perfect square on the left side: (x + 0.0002065)(x + 0.0002065) = 0.00054104264225 Calculate the square root of the right side: 0.023260323 Break this problem into two subproblems by setting (x + 0.0002065) equal to 0.023260323 and -0.023260323.Subproblem 1
x + 0.0002065 = 0.023260323 Simplifying x + 0.0002065 = 0.023260323 Reorder the terms: 0.0002065 + x = 0.023260323 Solving 0.0002065 + x = 0.023260323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0002065' to each side of the equation. 0.0002065 + -0.0002065 + x = 0.023260323 + -0.0002065 Combine like terms: 0.0002065 + -0.0002065 = 0.0000000 0.0000000 + x = 0.023260323 + -0.0002065 x = 0.023260323 + -0.0002065 Combine like terms: 0.023260323 + -0.0002065 = 0.023053823 x = 0.023053823 Simplifying x = 0.023053823Subproblem 2
x + 0.0002065 = -0.023260323 Simplifying x + 0.0002065 = -0.023260323 Reorder the terms: 0.0002065 + x = -0.023260323 Solving 0.0002065 + x = -0.023260323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0002065' to each side of the equation. 0.0002065 + -0.0002065 + x = -0.023260323 + -0.0002065 Combine like terms: 0.0002065 + -0.0002065 = 0.0000000 0.0000000 + x = -0.023260323 + -0.0002065 x = -0.023260323 + -0.0002065 Combine like terms: -0.023260323 + -0.0002065 = -0.023466823 x = -0.023466823 Simplifying x = -0.023466823Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.023053823, -0.023466823}
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