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Simplifying 633.024 = 2 * 3.14r2 + 3.14r * 16.2 Multiply 2 * 3.14 633.024 = 6.28r2 + 3.14r * 16.2 Reorder the terms for easier multiplication: 633.024 = 6.28r2 + 3.14 * 16.2r Multiply 3.14 * 16.2 633.024 = 6.28r2 + 50.868r Reorder the terms: 633.024 = 50.868r + 6.28r2 Solving 633.024 = 50.868r + 6.28r2 Solving for variable 'r'. Reorder the terms: 633.024 + -50.868r + -6.28r2 = 50.868r + -50.868r + 6.28r2 + -6.28r2 Combine like terms: 50.868r + -50.868r = 0.000 633.024 + -50.868r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 633.024 + -50.868r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 633.024 + -50.868r + -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'. -100.8 + 8.1r + r2 = 0 Move the constant term to the right: Add '100.8' to each side of the equation. -100.8 + 8.1r + 100.8 + r2 = 0 + 100.8 Reorder the terms: -100.8 + 100.8 + 8.1r + r2 = 0 + 100.8 Combine like terms: -100.8 + 100.8 = 0.0 0.0 + 8.1r + r2 = 0 + 100.8 8.1r + r2 = 0 + 100.8 Combine like terms: 0 + 100.8 = 100.8 8.1r + r2 = 100.8 The r term is 8.1r. Take half its coefficient (4.05). Square it (16.4025) and add it to both sides. Add '16.4025' to each side of the equation. 8.1r + 16.4025 + r2 = 100.8 + 16.4025 Reorder the terms: 16.4025 + 8.1r + r2 = 100.8 + 16.4025 Combine like terms: 100.8 + 16.4025 = 117.2025 16.4025 + 8.1r + r2 = 117.2025 Factor a perfect square on the left side: (r + 4.05)(r + 4.05) = 117.2025 Calculate the square root of the right side: 10.826010345 Break this problem into two subproblems by setting (r + 4.05) equal to 10.826010345 and -10.826010345.Subproblem 1
r + 4.05 = 10.826010345 Simplifying r + 4.05 = 10.826010345 Reorder the terms: 4.05 + r = 10.826010345 Solving 4.05 + r = 10.826010345 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.05' to each side of the equation. 4.05 + -4.05 + r = 10.826010345 + -4.05 Combine like terms: 4.05 + -4.05 = 0.00 0.00 + r = 10.826010345 + -4.05 r = 10.826010345 + -4.05 Combine like terms: 10.826010345 + -4.05 = 6.776010345 r = 6.776010345 Simplifying r = 6.776010345Subproblem 2
r + 4.05 = -10.826010345 Simplifying r + 4.05 = -10.826010345 Reorder the terms: 4.05 + r = -10.826010345 Solving 4.05 + r = -10.826010345 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.05' to each side of the equation. 4.05 + -4.05 + r = -10.826010345 + -4.05 Combine like terms: 4.05 + -4.05 = 0.00 0.00 + r = -10.826010345 + -4.05 r = -10.826010345 + -4.05 Combine like terms: -10.826010345 + -4.05 = -14.876010345 r = -14.876010345 Simplifying r = -14.876010345Solution
The solution to the problem is based on the solutions from the subproblems. r = {6.776010345, -14.876010345}
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