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Simplifying 406.8184 = 2 * 3.14r2 + 2 * 3.14r * 16 Multiply 2 * 3.14 406.8184 = 6.28r2 + 2 * 3.14r * 16 Reorder the terms for easier multiplication: 406.8184 = 6.28r2 + 2 * 3.14 * 16r Multiply 2 * 3.14 406.8184 = 6.28r2 + 6.28 * 16r Multiply 6.28 * 16 406.8184 = 6.28r2 + 100.48r Reorder the terms: 406.8184 = 100.48r + 6.28r2 Solving 406.8184 = 100.48r + 6.28r2 Solving for variable 'r'. Reorder the terms: 406.8184 + -100.48r + -6.28r2 = 100.48r + -100.48r + 6.28r2 + -6.28r2 Combine like terms: 100.48r + -100.48r = 0.00 406.8184 + -100.48r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 406.8184 + -100.48r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 406.8184 + -100.48r + -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'. -64.78 + 16r + r2 = 0 Move the constant term to the right: Add '64.78' to each side of the equation. -64.78 + 16r + 64.78 + r2 = 0 + 64.78 Reorder the terms: -64.78 + 64.78 + 16r + r2 = 0 + 64.78 Combine like terms: -64.78 + 64.78 = 0.00 0.00 + 16r + r2 = 0 + 64.78 16r + r2 = 0 + 64.78 Combine like terms: 0 + 64.78 = 64.78 16r + r2 = 64.78 The r term is 16r. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16r + 64 + r2 = 64.78 + 64 Reorder the terms: 64 + 16r + r2 = 64.78 + 64 Combine like terms: 64.78 + 64 = 128.78 64 + 16r + r2 = 128.78 Factor a perfect square on the left side: (r + 8)(r + 8) = 128.78 Calculate the square root of the right side: 11.348127599 Break this problem into two subproblems by setting (r + 8) equal to 11.348127599 and -11.348127599.Subproblem 1
r + 8 = 11.348127599 Simplifying r + 8 = 11.348127599 Reorder the terms: 8 + r = 11.348127599 Solving 8 + r = 11.348127599 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + r = 11.348127599 + -8 Combine like terms: 8 + -8 = 0 0 + r = 11.348127599 + -8 r = 11.348127599 + -8 Combine like terms: 11.348127599 + -8 = 3.348127599 r = 3.348127599 Simplifying r = 3.348127599Subproblem 2
r + 8 = -11.348127599 Simplifying r + 8 = -11.348127599 Reorder the terms: 8 + r = -11.348127599 Solving 8 + r = -11.348127599 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + r = -11.348127599 + -8 Combine like terms: 8 + -8 = 0 0 + r = -11.348127599 + -8 r = -11.348127599 + -8 Combine like terms: -11.348127599 + -8 = -19.348127599 r = -19.348127599 Simplifying r = -19.348127599Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.348127599, -19.348127599}
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