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Simplifying 3040.2736 = 2r2 + 2 * 3.14r * 93.9 Reorder the terms for easier multiplication: 3040.2736 = 2r2 + 2 * 3.14 * 93.9r Multiply 2 * 3.14 3040.2736 = 2r2 + 6.28 * 93.9r Multiply 6.28 * 93.9 3040.2736 = 2r2 + 589.692r Reorder the terms: 3040.2736 = 589.692r + 2r2 Solving 3040.2736 = 589.692r + 2r2 Solving for variable 'r'. Reorder the terms: 3040.2736 + -589.692r + -2r2 = 589.692r + -589.692r + 2r2 + -2r2 Combine like terms: 589.692r + -589.692r = 0.000 3040.2736 + -589.692r + -2r2 = 0.000 + 2r2 + -2r2 3040.2736 + -589.692r + -2r2 = 2r2 + -2r2 Combine like terms: 2r2 + -2r2 = 0 3040.2736 + -589.692r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1520.1368 + 294.846r + r2 = 0 Move the constant term to the right: Add '1520.1368' to each side of the equation. -1520.1368 + 294.846r + 1520.1368 + r2 = 0 + 1520.1368 Reorder the terms: -1520.1368 + 1520.1368 + 294.846r + r2 = 0 + 1520.1368 Combine like terms: -1520.1368 + 1520.1368 = 0.0000 0.0000 + 294.846r + r2 = 0 + 1520.1368 294.846r + r2 = 0 + 1520.1368 Combine like terms: 0 + 1520.1368 = 1520.1368 294.846r + r2 = 1520.1368 The r term is 294.846r. Take half its coefficient (147.423). Square it (21733.54093) and add it to both sides. Add '21733.54093' to each side of the equation. 294.846r + 21733.54093 + r2 = 1520.1368 + 21733.54093 Reorder the terms: 21733.54093 + 294.846r + r2 = 1520.1368 + 21733.54093 Combine like terms: 1520.1368 + 21733.54093 = 23253.67773 21733.54093 + 294.846r + r2 = 23253.67773 Factor a perfect square on the left side: (r + 147.423)(r + 147.423) = 23253.67773 Calculate the square root of the right side: 152.491566095 Break this problem into two subproblems by setting (r + 147.423) equal to 152.491566095 and -152.491566095.Subproblem 1
r + 147.423 = 152.491566095 Simplifying r + 147.423 = 152.491566095 Reorder the terms: 147.423 + r = 152.491566095 Solving 147.423 + r = 152.491566095 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-147.423' to each side of the equation. 147.423 + -147.423 + r = 152.491566095 + -147.423 Combine like terms: 147.423 + -147.423 = 0.000 0.000 + r = 152.491566095 + -147.423 r = 152.491566095 + -147.423 Combine like terms: 152.491566095 + -147.423 = 5.068566095 r = 5.068566095 Simplifying r = 5.068566095Subproblem 2
r + 147.423 = -152.491566095 Simplifying r + 147.423 = -152.491566095 Reorder the terms: 147.423 + r = -152.491566095 Solving 147.423 + r = -152.491566095 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-147.423' to each side of the equation. 147.423 + -147.423 + r = -152.491566095 + -147.423 Combine like terms: 147.423 + -147.423 = 0.000 0.000 + r = -152.491566095 + -147.423 r = -152.491566095 + -147.423 Combine like terms: -152.491566095 + -147.423 = -299.914566095 r = -299.914566095 Simplifying r = -299.914566095Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.068566095, -299.914566095}
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