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Simplifying 756.74 = 2 * 3.14r2 + 2 * 3.14r * 23.5 Multiply 2 * 3.14 756.74 = 6.28r2 + 2 * 3.14r * 23.5 Reorder the terms for easier multiplication: 756.74 = 6.28r2 + 2 * 3.14 * 23.5r Multiply 2 * 3.14 756.74 = 6.28r2 + 6.28 * 23.5r Multiply 6.28 * 23.5 756.74 = 6.28r2 + 147.58r Reorder the terms: 756.74 = 147.58r + 6.28r2 Solving 756.74 = 147.58r + 6.28r2 Solving for variable 'r'. Reorder the terms: 756.74 + -147.58r + -6.28r2 = 147.58r + -147.58r + 6.28r2 + -6.28r2 Combine like terms: 147.58r + -147.58r = 0.00 756.74 + -147.58r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 756.74 + -147.58r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 756.74 + -147.58r + -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'. -120.5 + 23.5r + r2 = 0 Move the constant term to the right: Add '120.5' to each side of the equation. -120.5 + 23.5r + 120.5 + r2 = 0 + 120.5 Reorder the terms: -120.5 + 120.5 + 23.5r + r2 = 0 + 120.5 Combine like terms: -120.5 + 120.5 = 0.0 0.0 + 23.5r + r2 = 0 + 120.5 23.5r + r2 = 0 + 120.5 Combine like terms: 0 + 120.5 = 120.5 23.5r + r2 = 120.5 The r term is 23.5r. Take half its coefficient (11.75). Square it (138.0625) and add it to both sides. Add '138.0625' to each side of the equation. 23.5r + 138.0625 + r2 = 120.5 + 138.0625 Reorder the terms: 138.0625 + 23.5r + r2 = 120.5 + 138.0625 Combine like terms: 120.5 + 138.0625 = 258.5625 138.0625 + 23.5r + r2 = 258.5625 Factor a perfect square on the left side: (r + 11.75)(r + 11.75) = 258.5625 Calculate the square root of the right side: 16.079878731 Break this problem into two subproblems by setting (r + 11.75) equal to 16.079878731 and -16.079878731.Subproblem 1
r + 11.75 = 16.079878731 Simplifying r + 11.75 = 16.079878731 Reorder the terms: 11.75 + r = 16.079878731 Solving 11.75 + r = 16.079878731 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-11.75' to each side of the equation. 11.75 + -11.75 + r = 16.079878731 + -11.75 Combine like terms: 11.75 + -11.75 = 0.00 0.00 + r = 16.079878731 + -11.75 r = 16.079878731 + -11.75 Combine like terms: 16.079878731 + -11.75 = 4.329878731 r = 4.329878731 Simplifying r = 4.329878731Subproblem 2
r + 11.75 = -16.079878731 Simplifying r + 11.75 = -16.079878731 Reorder the terms: 11.75 + r = -16.079878731 Solving 11.75 + r = -16.079878731 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-11.75' to each side of the equation. 11.75 + -11.75 + r = -16.079878731 + -11.75 Combine like terms: 11.75 + -11.75 = 0.00 0.00 + r = -16.079878731 + -11.75 r = -16.079878731 + -11.75 Combine like terms: -16.079878731 + -11.75 = -27.829878731 r = -27.829878731 Simplifying r = -27.829878731Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.329878731, -27.829878731}
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