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Simplifying 2007.088 = 2 * 3.14r * r + 2 * 3.14r * 93.9 Multiply 2 * 3.14 2007.088 = 6.28r * r + 2 * 3.14r * 93.9 Multiply r * r 2007.088 = 6.28r2 + 2 * 3.14r * 93.9 Reorder the terms for easier multiplication: 2007.088 = 6.28r2 + 2 * 3.14 * 93.9r Multiply 2 * 3.14 2007.088 = 6.28r2 + 6.28 * 93.9r Multiply 6.28 * 93.9 2007.088 = 6.28r2 + 589.692r Reorder the terms: 2007.088 = 589.692r + 6.28r2 Solving 2007.088 = 589.692r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2007.088 + -589.692r + -6.28r2 = 589.692r + -589.692r + 6.28r2 + -6.28r2 Combine like terms: 589.692r + -589.692r = 0.000 2007.088 + -589.692r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 2007.088 + -589.692r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2007.088 + -589.692r + -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'. -319.6 + 93.9r + r2 = 0 Move the constant term to the right: Add '319.6' to each side of the equation. -319.6 + 93.9r + 319.6 + r2 = 0 + 319.6 Reorder the terms: -319.6 + 319.6 + 93.9r + r2 = 0 + 319.6 Combine like terms: -319.6 + 319.6 = 0.0 0.0 + 93.9r + r2 = 0 + 319.6 93.9r + r2 = 0 + 319.6 Combine like terms: 0 + 319.6 = 319.6 93.9r + r2 = 319.6 The r term is 93.9r. Take half its coefficient (46.95). Square it (2204.3025) and add it to both sides. Add '2204.3025' to each side of the equation. 93.9r + 2204.3025 + r2 = 319.6 + 2204.3025 Reorder the terms: 2204.3025 + 93.9r + r2 = 319.6 + 2204.3025 Combine like terms: 319.6 + 2204.3025 = 2523.9025 2204.3025 + 93.9r + r2 = 2523.9025 Factor a perfect square on the left side: (r + 46.95)(r + 46.95) = 2523.9025 Calculate the square root of the right side: 50.238456386 Break this problem into two subproblems by setting (r + 46.95) equal to 50.238456386 and -50.238456386.Subproblem 1
r + 46.95 = 50.238456386 Simplifying r + 46.95 = 50.238456386 Reorder the terms: 46.95 + r = 50.238456386 Solving 46.95 + r = 50.238456386 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = 50.238456386 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = 50.238456386 + -46.95 r = 50.238456386 + -46.95 Combine like terms: 50.238456386 + -46.95 = 3.288456386 r = 3.288456386 Simplifying r = 3.288456386Subproblem 2
r + 46.95 = -50.238456386 Simplifying r + 46.95 = -50.238456386 Reorder the terms: 46.95 + r = -50.238456386 Solving 46.95 + r = -50.238456386 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-46.95' to each side of the equation. 46.95 + -46.95 + r = -50.238456386 + -46.95 Combine like terms: 46.95 + -46.95 = 0.00 0.00 + r = -50.238456386 + -46.95 r = -50.238456386 + -46.95 Combine like terms: -50.238456386 + -46.95 = -97.188456386 r = -97.188456386 Simplifying r = -97.188456386Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.288456386, -97.188456386}
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