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Simplifying (x) = -0.1x2 + 200x + 10000 x = -0.1x2 + 200x + 10000 Reorder the terms: x = 10000 + 200x + -0.1x2 Solving x = 10000 + 200x + -0.1x2 Solving for variable 'x'. Reorder the terms: -10000 + x + -200x + 0.1x2 = 10000 + 200x + -0.1x2 + -10000 + -200x + 0.1x2 Combine like terms: x + -200x = -199x -10000 + -199x + 0.1x2 = 10000 + 200x + -0.1x2 + -10000 + -200x + 0.1x2 Reorder the terms: -10000 + -199x + 0.1x2 = 10000 + -10000 + 200x + -200x + -0.1x2 + 0.1x2 Combine like terms: 10000 + -10000 = 0 -10000 + -199x + 0.1x2 = 0 + 200x + -200x + -0.1x2 + 0.1x2 -10000 + -199x + 0.1x2 = 200x + -200x + -0.1x2 + 0.1x2 Combine like terms: 200x + -200x = 0 -10000 + -199x + 0.1x2 = 0 + -0.1x2 + 0.1x2 -10000 + -199x + 0.1x2 = -0.1x2 + 0.1x2 Combine like terms: -0.1x2 + 0.1x2 = 0.0 -10000 + -199x + 0.1x2 = 0.0 Begin completing the square. Divide all terms by 0.1 the coefficient of the squared term: Divide each side by '0.1'. -100000 + -1990x + x2 = 0 Move the constant term to the right: Add '100000' to each side of the equation. -100000 + -1990x + 100000 + x2 = 0 + 100000 Reorder the terms: -100000 + 100000 + -1990x + x2 = 0 + 100000 Combine like terms: -100000 + 100000 = 0 0 + -1990x + x2 = 0 + 100000 -1990x + x2 = 0 + 100000 Combine like terms: 0 + 100000 = 100000 -1990x + x2 = 100000 The x term is -1990x. Take half its coefficient (-995). Square it (990025) and add it to both sides. Add '990025' to each side of the equation. -1990x + 990025 + x2 = 100000 + 990025 Reorder the terms: 990025 + -1990x + x2 = 100000 + 990025 Combine like terms: 100000 + 990025 = 1090025 990025 + -1990x + x2 = 1090025 Factor a perfect square on the left side: (x + -995)(x + -995) = 1090025 Calculate the square root of the right side: 1044.042623651 Break this problem into two subproblems by setting (x + -995) equal to 1044.042623651 and -1044.042623651.Subproblem 1
x + -995 = 1044.042623651 Simplifying x + -995 = 1044.042623651 Reorder the terms: -995 + x = 1044.042623651 Solving -995 + x = 1044.042623651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '995' to each side of the equation. -995 + 995 + x = 1044.042623651 + 995 Combine like terms: -995 + 995 = 0 0 + x = 1044.042623651 + 995 x = 1044.042623651 + 995 Combine like terms: 1044.042623651 + 995 = 2039.042623651 x = 2039.042623651 Simplifying x = 2039.042623651Subproblem 2
x + -995 = -1044.042623651 Simplifying x + -995 = -1044.042623651 Reorder the terms: -995 + x = -1044.042623651 Solving -995 + x = -1044.042623651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '995' to each side of the equation. -995 + 995 + x = -1044.042623651 + 995 Combine like terms: -995 + 995 = 0 0 + x = -1044.042623651 + 995 x = -1044.042623651 + 995 Combine like terms: -1044.042623651 + 995 = -49.042623651 x = -49.042623651 Simplifying x = -49.042623651Solution
The solution to the problem is based on the solutions from the subproblems. x = {2039.042623651, -49.042623651}
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