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Simplifying 200 = -1000x2 + 1100x + 2.5 Reorder the terms: 200 = 2.5 + 1100x + -1000x2 Solving 200 = 2.5 + 1100x + -1000x2 Solving for variable 'x'. Combine like terms: 200 + -2.5 = 197.5 197.5 + -1100x + 1000x2 = 2.5 + 1100x + -1000x2 + -2.5 + -1100x + 1000x2 Reorder the terms: 197.5 + -1100x + 1000x2 = 2.5 + -2.5 + 1100x + -1100x + -1000x2 + 1000x2 Combine like terms: 2.5 + -2.5 = 0.0 197.5 + -1100x + 1000x2 = 0.0 + 1100x + -1100x + -1000x2 + 1000x2 197.5 + -1100x + 1000x2 = 1100x + -1100x + -1000x2 + 1000x2 Combine like terms: 1100x + -1100x = 0 197.5 + -1100x + 1000x2 = 0 + -1000x2 + 1000x2 197.5 + -1100x + 1000x2 = -1000x2 + 1000x2 Combine like terms: -1000x2 + 1000x2 = 0 197.5 + -1100x + 1000x2 = 0 Begin completing the square. Divide all terms by 1000 the coefficient of the squared term: Divide each side by '1000'. 0.1975 + -1.1x + x2 = 0 Move the constant term to the right: Add '-0.1975' to each side of the equation. 0.1975 + -1.1x + -0.1975 + x2 = 0 + -0.1975 Reorder the terms: 0.1975 + -0.1975 + -1.1x + x2 = 0 + -0.1975 Combine like terms: 0.1975 + -0.1975 = 0.0000 0.0000 + -1.1x + x2 = 0 + -0.1975 -1.1x + x2 = 0 + -0.1975 Combine like terms: 0 + -0.1975 = -0.1975 -1.1x + x2 = -0.1975 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 = -0.1975 + 0.3025 Reorder the terms: 0.3025 + -1.1x + x2 = -0.1975 + 0.3025 Combine like terms: -0.1975 + 0.3025 = 0.105 0.3025 + -1.1x + x2 = 0.105 Factor a perfect square on the left side: (x + -0.55)(x + -0.55) = 0.105 Calculate the square root of the right side: 0.324037035 Break this problem into two subproblems by setting (x + -0.55) equal to 0.324037035 and -0.324037035.Subproblem 1
x + -0.55 = 0.324037035 Simplifying x + -0.55 = 0.324037035 Reorder the terms: -0.55 + x = 0.324037035 Solving -0.55 + x = 0.324037035 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 = 0.324037035 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = 0.324037035 + 0.55 x = 0.324037035 + 0.55 Combine like terms: 0.324037035 + 0.55 = 0.874037035 x = 0.874037035 Simplifying x = 0.874037035Subproblem 2
x + -0.55 = -0.324037035 Simplifying x + -0.55 = -0.324037035 Reorder the terms: -0.55 + x = -0.324037035 Solving -0.55 + x = -0.324037035 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 = -0.324037035 + 0.55 Combine like terms: -0.55 + 0.55 = 0.00 0.00 + x = -0.324037035 + 0.55 x = -0.324037035 + 0.55 Combine like terms: -0.324037035 + 0.55 = 0.225962965 x = 0.225962965 Simplifying x = 0.225962965Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.874037035, 0.225962965}
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