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Simplifying 0 = 6.28r2 + 1224.6r + -16981.12 Reorder the terms: 0 = -16981.12 + 1224.6r + 6.28r2 Solving 0 = -16981.12 + 1224.6r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 16981.12 = 16981.12 16981.12 + -1224.6r + -6.28r2 = -16981.12 + 1224.6r + 6.28r2 + 16981.12 + -1224.6r + -6.28r2 Reorder the terms: 16981.12 + -1224.6r + -6.28r2 = -16981.12 + 16981.12 + 1224.6r + -1224.6r + 6.28r2 + -6.28r2 Combine like terms: -16981.12 + 16981.12 = 0.00 16981.12 + -1224.6r + -6.28r2 = 0.00 + 1224.6r + -1224.6r + 6.28r2 + -6.28r2 16981.12 + -1224.6r + -6.28r2 = 1224.6r + -1224.6r + 6.28r2 + -6.28r2 Combine like terms: 1224.6r + -1224.6r = 0.0 16981.12 + -1224.6r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 16981.12 + -1224.6r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 16981.12 + -1224.6r + -6.28r2 = 0.00 Begin completing the square. Divide all terms by -6.28 the coefficient of the squared term: Divide each side by '-6.28'. -2704 + 195r + r2 = 0 Move the constant term to the right: Add '2704' to each side of the equation. -2704 + 195r + 2704 + r2 = 0 + 2704 Reorder the terms: -2704 + 2704 + 195r + r2 = 0 + 2704 Combine like terms: -2704 + 2704 = 0 0 + 195r + r2 = 0 + 2704 195r + r2 = 0 + 2704 Combine like terms: 0 + 2704 = 2704 195r + r2 = 2704 The r term is 195r. Take half its coefficient (97.5). Square it (9506.25) and add it to both sides. Add '9506.25' to each side of the equation. 195r + 9506.25 + r2 = 2704 + 9506.25 Reorder the terms: 9506.25 + 195r + r2 = 2704 + 9506.25 Combine like terms: 2704 + 9506.25 = 12210.25 9506.25 + 195r + r2 = 12210.25 Factor a perfect square on the left side: (r + 97.5)(r + 97.5) = 12210.25 Calculate the square root of the right side: 110.5 Break this problem into two subproblems by setting (r + 97.5) equal to 110.5 and -110.5.Subproblem 1
r + 97.5 = 110.5 Simplifying r + 97.5 = 110.5 Reorder the terms: 97.5 + r = 110.5 Solving 97.5 + r = 110.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-97.5' to each side of the equation. 97.5 + -97.5 + r = 110.5 + -97.5 Combine like terms: 97.5 + -97.5 = 0.0 0.0 + r = 110.5 + -97.5 r = 110.5 + -97.5 Combine like terms: 110.5 + -97.5 = 13 r = 13 Simplifying r = 13Subproblem 2
r + 97.5 = -110.5 Simplifying r + 97.5 = -110.5 Reorder the terms: 97.5 + r = -110.5 Solving 97.5 + r = -110.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-97.5' to each side of the equation. 97.5 + -97.5 + r = -110.5 + -97.5 Combine like terms: 97.5 + -97.5 = 0.0 0.0 + r = -110.5 + -97.5 r = -110.5 + -97.5 Combine like terms: -110.5 + -97.5 = -208 r = -208 Simplifying r = -208Solution
The solution to the problem is based on the solutions from the subproblems. r = {13, -208}
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