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Simplifying 47533.32 = 2 * 3.14r2 + 2 * 3.14r * 3.1 Multiply 2 * 3.14 47533.32 = 6.28r2 + 2 * 3.14r * 3.1 Reorder the terms for easier multiplication: 47533.32 = 6.28r2 + 2 * 3.14 * 3.1r Multiply 2 * 3.14 47533.32 = 6.28r2 + 6.28 * 3.1r Multiply 6.28 * 3.1 47533.32 = 6.28r2 + 19.468r Reorder the terms: 47533.32 = 19.468r + 6.28r2 Solving 47533.32 = 19.468r + 6.28r2 Solving for variable 'r'. Reorder the terms: 47533.32 + -19.468r + -6.28r2 = 19.468r + -19.468r + 6.28r2 + -6.28r2 Combine like terms: 19.468r + -19.468r = 0.000 47533.32 + -19.468r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 47533.32 + -19.468r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 47533.32 + -19.468r + -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'. -7569 + 3.1r + r2 = 0 Move the constant term to the right: Add '7569' to each side of the equation. -7569 + 3.1r + 7569 + r2 = 0 + 7569 Reorder the terms: -7569 + 7569 + 3.1r + r2 = 0 + 7569 Combine like terms: -7569 + 7569 = 0 0 + 3.1r + r2 = 0 + 7569 3.1r + r2 = 0 + 7569 Combine like terms: 0 + 7569 = 7569 3.1r + r2 = 7569 The r term is 3.1r. Take half its coefficient (1.55). Square it (2.4025) and add it to both sides. Add '2.4025' to each side of the equation. 3.1r + 2.4025 + r2 = 7569 + 2.4025 Reorder the terms: 2.4025 + 3.1r + r2 = 7569 + 2.4025 Combine like terms: 7569 + 2.4025 = 7571.4025 2.4025 + 3.1r + r2 = 7571.4025 Factor a perfect square on the left side: (r + 1.55)(r + 1.55) = 7571.4025 Calculate the square root of the right side: 87.013806376 Break this problem into two subproblems by setting (r + 1.55) equal to 87.013806376 and -87.013806376.Subproblem 1
r + 1.55 = 87.013806376 Simplifying r + 1.55 = 87.013806376 Reorder the terms: 1.55 + r = 87.013806376 Solving 1.55 + r = 87.013806376 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + r = 87.013806376 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + r = 87.013806376 + -1.55 r = 87.013806376 + -1.55 Combine like terms: 87.013806376 + -1.55 = 85.463806376 r = 85.463806376 Simplifying r = 85.463806376Subproblem 2
r + 1.55 = -87.013806376 Simplifying r + 1.55 = -87.013806376 Reorder the terms: 1.55 + r = -87.013806376 Solving 1.55 + r = -87.013806376 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.55' to each side of the equation. 1.55 + -1.55 + r = -87.013806376 + -1.55 Combine like terms: 1.55 + -1.55 = 0.00 0.00 + r = -87.013806376 + -1.55 r = -87.013806376 + -1.55 Combine like terms: -87.013806376 + -1.55 = -88.563806376 r = -88.563806376 Simplifying r = -88.563806376Solution
The solution to the problem is based on the solutions from the subproblems. r = {85.463806376, -88.563806376}
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