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Simplifying 202.5 = -1000x2 + 1100x Reorder the terms: 202.5 = 1100x + -1000x2 Solving 202.5 = 1100x + -1000x2 Solving for variable 'x'. Reorder the terms: 202.5 + -1100x + 1000x2 = 1100x + -1100x + -1000x2 + 1000x2 Combine like terms: 1100x + -1100x = 0 202.5 + -1100x + 1000x2 = 0 + -1000x2 + 1000x2 202.5 + -1100x + 1000x2 = -1000x2 + 1000x2 Combine like terms: -1000x2 + 1000x2 = 0 202.5 + -1100x + 1000x2 = 0 Begin completing the square. Divide all terms by 1000 the coefficient of the squared term: Divide each side by '1000'. 0.2025 + -1.1x + x2 = 0 Move the constant term to the right: Add '-0.2025' to each side of the equation. 0.2025 + -1.1x + -0.2025 + x2 = 0 + -0.2025 Reorder the terms: 0.2025 + -0.2025 + -1.1x + x2 = 0 + -0.2025 Combine like terms: 0.2025 + -0.2025 = 0.0000 0.0000 + -1.1x + x2 = 0 + -0.2025 -1.1x + x2 = 0 + -0.2025 Combine like terms: 0 + -0.2025 = -0.2025 -1.1x + x2 = -0.2025 The x term is -1.1x. Take half its coefficient (-0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. -1.1x + 0.3025 + x2 = -0.2025 + 0.3025 Reorder the terms: 0.3025 + -1.1x + x2 = -0.2025 + 0.3025 Combine like terms: -0.2025 + 0.3025 = 0.1 0.3025 + -1.1x + x2 = 0.1 Factor a perfect square on the left side: (x + -0.55)(x + -0.55) = 0.1 Calculate the square root of the right side: 0.316227766 Break this problem into two subproblems by setting (x + -0.55) equal to 0.316227766 and -0.316227766.Subproblem 1
x + -0.55 = 0.316227766 Simplifying x + -0.55 = 0.316227766 Reorder the terms: -0.55 + x = 0.316227766 Solving -0.55 + x = 0.316227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + x = 0.316227766 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = 0.316227766 + 0.55 x = 0.316227766 + 0.55 Combine like terms: 0.316227766 + 0.55 = 0.866227766 x = 0.866227766 Simplifying x = 0.866227766Subproblem 2
x + -0.55 = -0.316227766 Simplifying x + -0.55 = -0.316227766 Reorder the terms: -0.55 + x = -0.316227766 Solving -0.55 + x = -0.316227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.55' to each side of the equation. -0.55 + 0.55 + x = -0.316227766 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = -0.316227766 + 0.55 x = -0.316227766 + 0.55 Combine like terms: -0.316227766 + 0.55 = 0.233772234 x = 0.233772234 Simplifying x = 0.233772234Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.866227766, 0.233772234}
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