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Simplifying 621.72 = 6.28r2 + 6.28r(18.1) Reorder the terms for easier multiplication: 621.72 = 6.28r2 + 6.28 * 18.1r Multiply 6.28 * 18.1 621.72 = 6.28r2 + 113.668r Reorder the terms: 621.72 = 113.668r + 6.28r2 Solving 621.72 = 113.668r + 6.28r2 Solving for variable 'r'. Reorder the terms: 621.72 + -113.668r + -6.28r2 = 113.668r + -113.668r + 6.28r2 + -6.28r2 Combine like terms: 113.668r + -113.668r = 0.000 621.72 + -113.668r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 621.72 + -113.668r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 621.72 + -113.668r + -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'. -99 + 18.1r + r2 = 0 Move the constant term to the right: Add '99' to each side of the equation. -99 + 18.1r + 99 + r2 = 0 + 99 Reorder the terms: -99 + 99 + 18.1r + r2 = 0 + 99 Combine like terms: -99 + 99 = 0 0 + 18.1r + r2 = 0 + 99 18.1r + r2 = 0 + 99 Combine like terms: 0 + 99 = 99 18.1r + r2 = 99 The r term is 18.1r. Take half its coefficient (9.05). Square it (81.9025) and add it to both sides. Add '81.9025' to each side of the equation. 18.1r + 81.9025 + r2 = 99 + 81.9025 Reorder the terms: 81.9025 + 18.1r + r2 = 99 + 81.9025 Combine like terms: 99 + 81.9025 = 180.9025 81.9025 + 18.1r + r2 = 180.9025 Factor a perfect square on the left side: (r + 9.05)(r + 9.05) = 180.9025 Calculate the square root of the right side: 13.45 Break this problem into two subproblems by setting (r + 9.05) equal to 13.45 and -13.45.Subproblem 1
r + 9.05 = 13.45 Simplifying r + 9.05 = 13.45 Reorder the terms: 9.05 + r = 13.45 Solving 9.05 + r = 13.45 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.05' to each side of the equation. 9.05 + -9.05 + r = 13.45 + -9.05 Combine like terms: 9.05 + -9.05 = 0.00 0.00 + r = 13.45 + -9.05 r = 13.45 + -9.05 Combine like terms: 13.45 + -9.05 = 4.4 r = 4.4 Simplifying r = 4.4Subproblem 2
r + 9.05 = -13.45 Simplifying r + 9.05 = -13.45 Reorder the terms: 9.05 + r = -13.45 Solving 9.05 + r = -13.45 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.05' to each side of the equation. 9.05 + -9.05 + r = -13.45 + -9.05 Combine like terms: 9.05 + -9.05 = 0.00 0.00 + r = -13.45 + -9.05 r = -13.45 + -9.05 Combine like terms: -13.45 + -9.05 = -22.5 r = -22.5 Simplifying r = -22.5Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.4, -22.5}
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