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Simplifying x = 0.001x2 + 2.9x + -1000 Reorder the terms: x = -1000 + 2.9x + 0.001x2 Solving x = -1000 + 2.9x + 0.001x2 Solving for variable 'x'. Reorder the terms: 1000 + x + -2.9x + -0.001x2 = -1000 + 2.9x + 0.001x2 + 1000 + -2.9x + -0.001x2 Combine like terms: x + -2.9x = -1.9x 1000 + -1.9x + -0.001x2 = -1000 + 2.9x + 0.001x2 + 1000 + -2.9x + -0.001x2 Reorder the terms: 1000 + -1.9x + -0.001x2 = -1000 + 1000 + 2.9x + -2.9x + 0.001x2 + -0.001x2 Combine like terms: -1000 + 1000 = 0 1000 + -1.9x + -0.001x2 = 0 + 2.9x + -2.9x + 0.001x2 + -0.001x2 1000 + -1.9x + -0.001x2 = 2.9x + -2.9x + 0.001x2 + -0.001x2 Combine like terms: 2.9x + -2.9x = 0.0 1000 + -1.9x + -0.001x2 = 0.0 + 0.001x2 + -0.001x2 1000 + -1.9x + -0.001x2 = 0.001x2 + -0.001x2 Combine like terms: 0.001x2 + -0.001x2 = 0.000 1000 + -1.9x + -0.001x2 = 0.000 Begin completing the square. Divide all terms by -0.001 the coefficient of the squared term: Divide each side by '-0.001'. -1000000 + 1900x + x2 = 0 Move the constant term to the right: Add '1000000' to each side of the equation. -1000000 + 1900x + 1000000 + x2 = 0 + 1000000 Reorder the terms: -1000000 + 1000000 + 1900x + x2 = 0 + 1000000 Combine like terms: -1000000 + 1000000 = 0 0 + 1900x + x2 = 0 + 1000000 1900x + x2 = 0 + 1000000 Combine like terms: 0 + 1000000 = 1000000 1900x + x2 = 1000000 The x term is 1900x. Take half its coefficient (950). Square it (902500) and add it to both sides. Add '902500' to each side of the equation. 1900x + 902500 + x2 = 1000000 + 902500 Reorder the terms: 902500 + 1900x + x2 = 1000000 + 902500 Combine like terms: 1000000 + 902500 = 1902500 902500 + 1900x + x2 = 1902500 Factor a perfect square on the left side: (x + 950)(x + 950) = 1902500 Calculate the square root of the right side: 1379.311422413 Break this problem into two subproblems by setting (x + 950) equal to 1379.311422413 and -1379.311422413.Subproblem 1
x + 950 = 1379.311422413 Simplifying x + 950 = 1379.311422413 Reorder the terms: 950 + x = 1379.311422413 Solving 950 + x = 1379.311422413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-950' to each side of the equation. 950 + -950 + x = 1379.311422413 + -950 Combine like terms: 950 + -950 = 0 0 + x = 1379.311422413 + -950 x = 1379.311422413 + -950 Combine like terms: 1379.311422413 + -950 = 429.311422413 x = 429.311422413 Simplifying x = 429.311422413Subproblem 2
x + 950 = -1379.311422413 Simplifying x + 950 = -1379.311422413 Reorder the terms: 950 + x = -1379.311422413 Solving 950 + x = -1379.311422413 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-950' to each side of the equation. 950 + -950 + x = -1379.311422413 + -950 Combine like terms: 950 + -950 = 0 0 + x = -1379.311422413 + -950 x = -1379.311422413 + -950 Combine like terms: -1379.311422413 + -950 = -2329.311422413 x = -2329.311422413 Simplifying x = -2329.311422413Solution
The solution to the problem is based on the solutions from the subproblems. x = {429.311422413, -2329.311422413}
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