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Simplifying 0 = 6.28r2 + 137.532r + -1270.444 Reorder the terms: 0 = -1270.444 + 137.532r + 6.28r2 Solving 0 = -1270.444 + 137.532r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 1270.444 = 1270.444 1270.444 + -137.532r + -6.28r2 = -1270.444 + 137.532r + 6.28r2 + 1270.444 + -137.532r + -6.28r2 Reorder the terms: 1270.444 + -137.532r + -6.28r2 = -1270.444 + 1270.444 + 137.532r + -137.532r + 6.28r2 + -6.28r2 Combine like terms: -1270.444 + 1270.444 = 0.000 1270.444 + -137.532r + -6.28r2 = 0.000 + 137.532r + -137.532r + 6.28r2 + -6.28r2 1270.444 + -137.532r + -6.28r2 = 137.532r + -137.532r + 6.28r2 + -6.28r2 Combine like terms: 137.532r + -137.532r = 0.000 1270.444 + -137.532r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 1270.444 + -137.532r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 1270.444 + -137.532r + -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'. -202.3 + 21.9r + r2 = 0 Move the constant term to the right: Add '202.3' to each side of the equation. -202.3 + 21.9r + 202.3 + r2 = 0 + 202.3 Reorder the terms: -202.3 + 202.3 + 21.9r + r2 = 0 + 202.3 Combine like terms: -202.3 + 202.3 = 0.0 0.0 + 21.9r + r2 = 0 + 202.3 21.9r + r2 = 0 + 202.3 Combine like terms: 0 + 202.3 = 202.3 21.9r + r2 = 202.3 The r term is 21.9r. Take half its coefficient (10.95). Square it (119.9025) and add it to both sides. Add '119.9025' to each side of the equation. 21.9r + 119.9025 + r2 = 202.3 + 119.9025 Reorder the terms: 119.9025 + 21.9r + r2 = 202.3 + 119.9025 Combine like terms: 202.3 + 119.9025 = 322.2025 119.9025 + 21.9r + r2 = 322.2025 Factor a perfect square on the left side: (r + 10.95)(r + 10.95) = 322.2025 Calculate the square root of the right side: 17.95 Break this problem into two subproblems by setting (r + 10.95) equal to 17.95 and -17.95.Subproblem 1
r + 10.95 = 17.95 Simplifying r + 10.95 = 17.95 Reorder the terms: 10.95 + r = 17.95 Solving 10.95 + r = 17.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-10.95' to each side of the equation. 10.95 + -10.95 + r = 17.95 + -10.95 Combine like terms: 10.95 + -10.95 = 0.00 0.00 + r = 17.95 + -10.95 r = 17.95 + -10.95 Combine like terms: 17.95 + -10.95 = 7 r = 7 Simplifying r = 7Subproblem 2
r + 10.95 = -17.95 Simplifying r + 10.95 = -17.95 Reorder the terms: 10.95 + r = -17.95 Solving 10.95 + r = -17.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-10.95' to each side of the equation. 10.95 + -10.95 + r = -17.95 + -10.95 Combine like terms: 10.95 + -10.95 = 0.00 0.00 + r = -17.95 + -10.95 r = -17.95 + -10.95 Combine like terms: -17.95 + -10.95 = -28.9 r = -28.9 Simplifying r = -28.9Solution
The solution to the problem is based on the solutions from the subproblems. r = {7, -28.9}
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