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Simplifying 0 = 6.28r2 + 98.596r + -509.8104 Reorder the terms: 0 = -509.8104 + 98.596r + 6.28r2 Solving 0 = -509.8104 + 98.596r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 509.8104 = 509.8104 509.8104 + -98.596r + -6.28r2 = -509.8104 + 98.596r + 6.28r2 + 509.8104 + -98.596r + -6.28r2 Reorder the terms: 509.8104 + -98.596r + -6.28r2 = -509.8104 + 509.8104 + 98.596r + -98.596r + 6.28r2 + -6.28r2 Combine like terms: -509.8104 + 509.8104 = 0.0000 509.8104 + -98.596r + -6.28r2 = 0.0000 + 98.596r + -98.596r + 6.28r2 + -6.28r2 509.8104 + -98.596r + -6.28r2 = 98.596r + -98.596r + 6.28r2 + -6.28r2 Combine like terms: 98.596r + -98.596r = 0.000 509.8104 + -98.596r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 509.8104 + -98.596r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 509.8104 + -98.596r + -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'. -81.18 + 15.7r + r2 = 0 Move the constant term to the right: Add '81.18' to each side of the equation. -81.18 + 15.7r + 81.18 + r2 = 0 + 81.18 Reorder the terms: -81.18 + 81.18 + 15.7r + r2 = 0 + 81.18 Combine like terms: -81.18 + 81.18 = 0.00 0.00 + 15.7r + r2 = 0 + 81.18 15.7r + r2 = 0 + 81.18 Combine like terms: 0 + 81.18 = 81.18 15.7r + r2 = 81.18 The r term is 15.7r. Take half its coefficient (7.85). Square it (61.6225) and add it to both sides. Add '61.6225' to each side of the equation. 15.7r + 61.6225 + r2 = 81.18 + 61.6225 Reorder the terms: 61.6225 + 15.7r + r2 = 81.18 + 61.6225 Combine like terms: 81.18 + 61.6225 = 142.8025 61.6225 + 15.7r + r2 = 142.8025 Factor a perfect square on the left side: (r + 7.85)(r + 7.85) = 142.8025 Calculate the square root of the right side: 11.95 Break this problem into two subproblems by setting (r + 7.85) equal to 11.95 and -11.95.Subproblem 1
r + 7.85 = 11.95 Simplifying r + 7.85 = 11.95 Reorder the terms: 7.85 + r = 11.95 Solving 7.85 + r = 11.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.85' to each side of the equation. 7.85 + -7.85 + r = 11.95 + -7.85 Combine like terms: 7.85 + -7.85 = 0.00 0.00 + r = 11.95 + -7.85 r = 11.95 + -7.85 Combine like terms: 11.95 + -7.85 = 4.1 r = 4.1 Simplifying r = 4.1Subproblem 2
r + 7.85 = -11.95 Simplifying r + 7.85 = -11.95 Reorder the terms: 7.85 + r = -11.95 Solving 7.85 + r = -11.95 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.85' to each side of the equation. 7.85 + -7.85 + r = -11.95 + -7.85 Combine like terms: 7.85 + -7.85 = 0.00 0.00 + r = -11.95 + -7.85 r = -11.95 + -7.85 Combine like terms: -11.95 + -7.85 = -19.8 r = -19.8 Simplifying r = -19.8Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.1, -19.8}
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