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Simplifying 0 = 6.28r2 + 589.692r + -3040.2736 Reorder the terms: 0 = -3040.2736 + 589.692r + 6.28r2 Solving 0 = -3040.2736 + 589.692r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 3040.2736 = 3040.2736 3040.2736 + -589.692r + -6.28r2 = -3040.2736 + 589.692r + 6.28r2 + 3040.2736 + -589.692r + -6.28r2 Reorder the terms: 3040.2736 + -589.692r + -6.28r2 = -3040.2736 + 3040.2736 + 589.692r + -589.692r + 6.28r2 + -6.28r2 Combine like terms: -3040.2736 + 3040.2736 = 0.0000 3040.2736 + -589.692r + -6.28r2 = 0.0000 + 589.692r + -589.692r + 6.28r2 + -6.28r2 3040.2736 + -589.692r + -6.28r2 = 589.692r + -589.692r + 6.28r2 + -6.28r2 Combine like terms: 589.692r + -589.692r = 0.000 3040.2736 + -589.692r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 3040.2736 + -589.692r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 3040.2736 + -589.692r + -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'. -484.12 + 93.9r + r2 = 0 Move the constant term to the right: Add '484.12' to each side of the equation. -484.12 + 93.9r + 484.12 + r2 = 0 + 484.12 Reorder the terms: -484.12 + 484.12 + 93.9r + r2 = 0 + 484.12 Combine like terms: -484.12 + 484.12 = 0.00 0.00 + 93.9r + r2 = 0 + 484.12 93.9r + r2 = 0 + 484.12 Combine like terms: 0 + 484.12 = 484.12 93.9r + r2 = 484.12 The r term is 93.9r. Take half its coefficient (46.95). Square it (2204.3025) and add it to both sides. Add '2204.3025' to each side of the equation. 93.9r + 2204.3025 + r2 = 484.12 + 2204.3025 Reorder the terms: 2204.3025 + 93.9r + r2 = 484.12 + 2204.3025 Combine like terms: 484.12 + 2204.3025 = 2688.4225 2204.3025 + 93.9r + r2 = 2688.4225 Factor a perfect square on the left side: (r + 46.95)(r + 46.95) = 2688.4225 Calculate the square root of the right side: 51.85 Break this problem into two subproblems by setting (r + 46.95) equal to 51.85 and -51.85.Subproblem 1
r + 46.95 = 51.85 Simplifying r + 46.95 = 51.85 Reorder the terms: 46.95 + r = 51.85 Solving 46.95 + r = 51.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = 51.85 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = 51.85 + -46.95 r = 51.85 + -46.95 Combine like terms: 51.85 + -46.95 = 4.9 r = 4.9 Simplifying r = 4.9Subproblem 2
r + 46.95 = -51.85 Simplifying r + 46.95 = -51.85 Reorder the terms: 46.95 + r = -51.85 Solving 46.95 + r = -51.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = -51.85 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = -51.85 + -46.95 r = -51.85 + -46.95 Combine like terms: -51.85 + -46.95 = -98.8 r = -98.8 Simplifying r = -98.8Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.9, -98.8}
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