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Simplifying 500x + 0.1x2 + 17764 = 1000x + -0.01x2 Reorder the terms: 17764 + 500x + 0.1x2 = 1000x + -0.01x2 Solving 17764 + 500x + 0.1x2 = 1000x + -0.01x2 Solving for variable 'x'. Reorder the terms: 17764 + 500x + -1000x + 0.1x2 + 0.01x2 = 1000x + -0.01x2 + -1000x + 0.01x2 Combine like terms: 500x + -1000x = -500x 17764 + -500x + 0.1x2 + 0.01x2 = 1000x + -0.01x2 + -1000x + 0.01x2 Combine like terms: 0.1x2 + 0.01x2 = 0.11x2 17764 + -500x + 0.11x2 = 1000x + -0.01x2 + -1000x + 0.01x2 Reorder the terms: 17764 + -500x + 0.11x2 = 1000x + -1000x + -0.01x2 + 0.01x2 Combine like terms: 1000x + -1000x = 0 17764 + -500x + 0.11x2 = 0 + -0.01x2 + 0.01x2 17764 + -500x + 0.11x2 = -0.01x2 + 0.01x2 Combine like terms: -0.01x2 + 0.01x2 = 0.00 17764 + -500x + 0.11x2 = 0.00 Begin completing the square. Divide all terms by 0.11 the coefficient of the squared term: Divide each side by '0.11'. 161490.9091 + -4545.454545x + x2 = 0 Move the constant term to the right: Add '-161490.9091' to each side of the equation. 161490.9091 + -4545.454545x + -161490.9091 + x2 = 0 + -161490.9091 Reorder the terms: 161490.9091 + -161490.9091 + -4545.454545x + x2 = 0 + -161490.9091 Combine like terms: 161490.9091 + -161490.9091 = 0.0000 0.0000 + -4545.454545x + x2 = 0 + -161490.9091 -4545.454545x + x2 = 0 + -161490.9091 Combine like terms: 0 + -161490.9091 = -161490.9091 -4545.454545x + x2 = -161490.9091 The x term is -4545.454545x. Take half its coefficient (-2272.727273). Square it (5165289.257) and add it to both sides. Add '5165289.257' to each side of the equation. -4545.454545x + 5165289.257 + x2 = -161490.9091 + 5165289.257 Reorder the terms: 5165289.257 + -4545.454545x + x2 = -161490.9091 + 5165289.257 Combine like terms: -161490.9091 + 5165289.257 = 5003798.3479 5165289.257 + -4545.454545x + x2 = 5003798.3479 Factor a perfect square on the left side: (x + -2272.727273)(x + -2272.727273) = 5003798.3479 Calculate the square root of the right side: 2236.917152668 Break this problem into two subproblems by setting (x + -2272.727273) equal to 2236.917152668 and -2236.917152668.Subproblem 1
x + -2272.727273 = 2236.917152668 Simplifying x + -2272.727273 = 2236.917152668 Reorder the terms: -2272.727273 + x = 2236.917152668 Solving -2272.727273 + x = 2236.917152668 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2272.727273' to each side of the equation. -2272.727273 + 2272.727273 + x = 2236.917152668 + 2272.727273 Combine like terms: -2272.727273 + 2272.727273 = 0.000000 0.000000 + x = 2236.917152668 + 2272.727273 x = 2236.917152668 + 2272.727273 Combine like terms: 2236.917152668 + 2272.727273 = 4509.644425668 x = 4509.644425668 Simplifying x = 4509.644425668Subproblem 2
x + -2272.727273 = -2236.917152668 Simplifying x + -2272.727273 = -2236.917152668 Reorder the terms: -2272.727273 + x = -2236.917152668 Solving -2272.727273 + x = -2236.917152668 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2272.727273' to each side of the equation. -2272.727273 + 2272.727273 + x = -2236.917152668 + 2272.727273 Combine like terms: -2272.727273 + 2272.727273 = 0.000000 0.000000 + x = -2236.917152668 + 2272.727273 x = -2236.917152668 + 2272.727273 Combine like terms: -2236.917152668 + 2272.727273 = 35.810120332 x = 35.810120332 Simplifying x = 35.810120332Solution
The solution to the problem is based on the solutions from the subproblems. x = {4509.644425668, 35.810120332}
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