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Simplifying 512.3854 = 2 * 3.14r2 + 2 * 3.14r * 15.8 Multiply 2 * 3.14 512.3854 = 6.28r2 + 2 * 3.14r * 15.8 Reorder the terms for easier multiplication: 512.3854 = 6.28r2 + 2 * 3.14 * 15.8r Multiply 2 * 3.14 512.3854 = 6.28r2 + 6.28 * 15.8r Multiply 6.28 * 15.8 512.3854 = 6.28r2 + 99.224r Reorder the terms: 512.3854 = 99.224r + 6.28r2 Solving 512.3854 = 99.224r + 6.28r2 Solving for variable 'r'. Reorder the terms: 512.3854 + -99.224r + -6.28r2 = 99.224r + -99.224r + 6.28r2 + -6.28r2 Combine like terms: 99.224r + -99.224r = 0.000 512.3854 + -99.224r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 512.3854 + -99.224r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 512.3854 + -99.224r + -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'. -81.59003185 + 15.8r + r2 = 0 Move the constant term to the right: Add '81.59003185' to each side of the equation. -81.59003185 + 15.8r + 81.59003185 + r2 = 0 + 81.59003185 Reorder the terms: -81.59003185 + 81.59003185 + 15.8r + r2 = 0 + 81.59003185 Combine like terms: -81.59003185 + 81.59003185 = 0.00000000 0.00000000 + 15.8r + r2 = 0 + 81.59003185 15.8r + r2 = 0 + 81.59003185 Combine like terms: 0 + 81.59003185 = 81.59003185 15.8r + r2 = 81.59003185 The r term is 15.8r. Take half its coefficient (7.9). Square it (62.41) and add it to both sides. Add '62.41' to each side of the equation. 15.8r + 62.41 + r2 = 81.59003185 + 62.41 Reorder the terms: 62.41 + 15.8r + r2 = 81.59003185 + 62.41 Combine like terms: 81.59003185 + 62.41 = 144.00003185 62.41 + 15.8r + r2 = 144.00003185 Factor a perfect square on the left side: (r + 7.9)(r + 7.9) = 144.00003185 Calculate the square root of the right side: 12.000001327 Break this problem into two subproblems by setting (r + 7.9) equal to 12.000001327 and -12.000001327.Subproblem 1
r + 7.9 = 12.000001327 Simplifying r + 7.9 = 12.000001327 Reorder the terms: 7.9 + r = 12.000001327 Solving 7.9 + r = 12.000001327 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.9' to each side of the equation. 7.9 + -7.9 + r = 12.000001327 + -7.9 Combine like terms: 7.9 + -7.9 = 0.0 0.0 + r = 12.000001327 + -7.9 r = 12.000001327 + -7.9 Combine like terms: 12.000001327 + -7.9 = 4.100001327 r = 4.100001327 Simplifying r = 4.100001327Subproblem 2
r + 7.9 = -12.000001327 Simplifying r + 7.9 = -12.000001327 Reorder the terms: 7.9 + r = -12.000001327 Solving 7.9 + r = -12.000001327 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-7.9' to each side of the equation. 7.9 + -7.9 + r = -12.000001327 + -7.9 Combine like terms: 7.9 + -7.9 = 0.0 0.0 + r = -12.000001327 + -7.9 r = -12.000001327 + -7.9 Combine like terms: -12.000001327 + -7.9 = -19.900001327 r = -19.900001327 Simplifying r = -19.900001327Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.100001327, -19.900001327}
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