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Simplifying 509.8104 = 2 * 3.14r2 + 2 * 3.14r * 55.6 Multiply 2 * 3.14 509.8104 = 6.28r2 + 2 * 3.14r * 55.6 Reorder the terms for easier multiplication: 509.8104 = 6.28r2 + 2 * 3.14 * 55.6r Multiply 2 * 3.14 509.8104 = 6.28r2 + 6.28 * 55.6r Multiply 6.28 * 55.6 509.8104 = 6.28r2 + 349.168r Reorder the terms: 509.8104 = 349.168r + 6.28r2 Solving 509.8104 = 349.168r + 6.28r2 Solving for variable 'r'. Reorder the terms: 509.8104 + -349.168r + -6.28r2 = 349.168r + -349.168r + 6.28r2 + -6.28r2 Combine like terms: 349.168r + -349.168r = 0.000 509.8104 + -349.168r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 509.8104 + -349.168r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 509.8104 + -349.168r + -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 + 55.6r + r2 = 0 Move the constant term to the right: Add '81.18' to each side of the equation. -81.18 + 55.6r + 81.18 + r2 = 0 + 81.18 Reorder the terms: -81.18 + 81.18 + 55.6r + r2 = 0 + 81.18 Combine like terms: -81.18 + 81.18 = 0.00 0.00 + 55.6r + r2 = 0 + 81.18 55.6r + r2 = 0 + 81.18 Combine like terms: 0 + 81.18 = 81.18 55.6r + r2 = 81.18 The r term is 55.6r. Take half its coefficient (27.8). Square it (772.84) and add it to both sides. Add '772.84' to each side of the equation. 55.6r + 772.84 + r2 = 81.18 + 772.84 Reorder the terms: 772.84 + 55.6r + r2 = 81.18 + 772.84 Combine like terms: 81.18 + 772.84 = 854.02 772.84 + 55.6r + r2 = 854.02 Factor a perfect square on the left side: (r + 27.8)(r + 27.8) = 854.02 Calculate the square root of the right side: 29.223620583 Break this problem into two subproblems by setting (r + 27.8) equal to 29.223620583 and -29.223620583.Subproblem 1
r + 27.8 = 29.223620583 Simplifying r + 27.8 = 29.223620583 Reorder the terms: 27.8 + r = 29.223620583 Solving 27.8 + r = 29.223620583 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-27.8' to each side of the equation. 27.8 + -27.8 + r = 29.223620583 + -27.8 Combine like terms: 27.8 + -27.8 = 0.0 0.0 + r = 29.223620583 + -27.8 r = 29.223620583 + -27.8 Combine like terms: 29.223620583 + -27.8 = 1.423620583 r = 1.423620583 Simplifying r = 1.423620583Subproblem 2
r + 27.8 = -29.223620583 Simplifying r + 27.8 = -29.223620583 Reorder the terms: 27.8 + r = -29.223620583 Solving 27.8 + r = -29.223620583 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-27.8' to each side of the equation. 27.8 + -27.8 + r = -29.223620583 + -27.8 Combine like terms: 27.8 + -27.8 = 0.0 0.0 + r = -29.223620583 + -27.8 r = -29.223620583 + -27.8 Combine like terms: -29.223620583 + -27.8 = -57.023620583 r = -57.023620583 Simplifying r = -57.023620583Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.423620583, -57.023620583}
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