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Simplifying 141.5512 = 2 * 3.14r * 7.5 * 7.5 + 2 * 3.14r2 Reorder the terms for easier multiplication: 141.5512 = 2 * 3.14 * 7.5 * 7.5r + 2 * 3.14r2 Multiply 2 * 3.14 141.5512 = 6.28 * 7.5 * 7.5r + 2 * 3.14r2 Multiply 6.28 * 7.5 141.5512 = 47.1 * 7.5r + 2 * 3.14r2 Multiply 47.1 * 7.5 141.5512 = 353.25r + 2 * 3.14r2 Multiply 2 * 3.14 141.5512 = 353.25r + 6.28r2 Solving 141.5512 = 353.25r + 6.28r2 Solving for variable 'r'. Reorder the terms: 141.5512 + -353.25r + -6.28r2 = 353.25r + -353.25r + 6.28r2 + -6.28r2 Combine like terms: 353.25r + -353.25r = 0.00 141.5512 + -353.25r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 141.5512 + -353.25r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 141.5512 + -353.25r + -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'. -22.54 + 56.25r + r2 = 0 Move the constant term to the right: Add '22.54' to each side of the equation. -22.54 + 56.25r + 22.54 + r2 = 0 + 22.54 Reorder the terms: -22.54 + 22.54 + 56.25r + r2 = 0 + 22.54 Combine like terms: -22.54 + 22.54 = 0.00 0.00 + 56.25r + r2 = 0 + 22.54 56.25r + r2 = 0 + 22.54 Combine like terms: 0 + 22.54 = 22.54 56.25r + r2 = 22.54 The r term is 56.25r. Take half its coefficient (28.125). Square it (791.015625) and add it to both sides. Add '791.015625' to each side of the equation. 56.25r + 791.015625 + r2 = 22.54 + 791.015625 Reorder the terms: 791.015625 + 56.25r + r2 = 22.54 + 791.015625 Combine like terms: 22.54 + 791.015625 = 813.555625 791.015625 + 56.25r + r2 = 813.555625 Factor a perfect square on the left side: (r + 28.125)(r + 28.125) = 813.555625 Calculate the square root of the right side: 28.522896504 Break this problem into two subproblems by setting (r + 28.125) equal to 28.522896504 and -28.522896504.Subproblem 1
r + 28.125 = 28.522896504 Simplifying r + 28.125 = 28.522896504 Reorder the terms: 28.125 + r = 28.522896504 Solving 28.125 + r = 28.522896504 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-28.125' to each side of the equation. 28.125 + -28.125 + r = 28.522896504 + -28.125 Combine like terms: 28.125 + -28.125 = 0.000 0.000 + r = 28.522896504 + -28.125 r = 28.522896504 + -28.125 Combine like terms: 28.522896504 + -28.125 = 0.397896504 r = 0.397896504 Simplifying r = 0.397896504Subproblem 2
r + 28.125 = -28.522896504 Simplifying r + 28.125 = -28.522896504 Reorder the terms: 28.125 + r = -28.522896504 Solving 28.125 + r = -28.522896504 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-28.125' to each side of the equation. 28.125 + -28.125 + r = -28.522896504 + -28.125 Combine like terms: 28.125 + -28.125 = 0.000 0.000 + r = -28.522896504 + -28.125 r = -28.522896504 + -28.125 Combine like terms: -28.522896504 + -28.125 = -56.647896504 r = -56.647896504 Simplifying r = -56.647896504Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.397896504, -56.647896504}
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