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Simplifying 2010.6048 = 6.28r2 + 6.28r * 36.5 Reorder the terms for easier multiplication: 2010.6048 = 6.28r2 + 6.28 * 36.5r Multiply 6.28 * 36.5 2010.6048 = 6.28r2 + 229.22r Reorder the terms: 2010.6048 = 229.22r + 6.28r2 Solving 2010.6048 = 229.22r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2010.6048 + -229.22r + -6.28r2 = 229.22r + -229.22r + 6.28r2 + -6.28r2 Combine like terms: 229.22r + -229.22r = 0.00 2010.6048 + -229.22r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 2010.6048 + -229.22r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2010.6048 + -229.22r + -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'. -320.16 + 36.5r + r2 = 0 Move the constant term to the right: Add '320.16' to each side of the equation. -320.16 + 36.5r + 320.16 + r2 = 0 + 320.16 Reorder the terms: -320.16 + 320.16 + 36.5r + r2 = 0 + 320.16 Combine like terms: -320.16 + 320.16 = 0.00 0.00 + 36.5r + r2 = 0 + 320.16 36.5r + r2 = 0 + 320.16 Combine like terms: 0 + 320.16 = 320.16 36.5r + r2 = 320.16 The r term is 36.5r. Take half its coefficient (18.25). Square it (333.0625) and add it to both sides. Add '333.0625' to each side of the equation. 36.5r + 333.0625 + r2 = 320.16 + 333.0625 Reorder the terms: 333.0625 + 36.5r + r2 = 320.16 + 333.0625 Combine like terms: 320.16 + 333.0625 = 653.2225 333.0625 + 36.5r + r2 = 653.2225 Factor a perfect square on the left side: (r + 18.25)(r + 18.25) = 653.2225 Calculate the square root of the right side: 25.558217856 Break this problem into two subproblems by setting (r + 18.25) equal to 25.558217856 and -25.558217856.Subproblem 1
r + 18.25 = 25.558217856 Simplifying r + 18.25 = 25.558217856 Reorder the terms: 18.25 + r = 25.558217856 Solving 18.25 + r = 25.558217856 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.25' to each side of the equation. 18.25 + -18.25 + r = 25.558217856 + -18.25 Combine like terms: 18.25 + -18.25 = 0.00 0.00 + r = 25.558217856 + -18.25 r = 25.558217856 + -18.25 Combine like terms: 25.558217856 + -18.25 = 7.308217856 r = 7.308217856 Simplifying r = 7.308217856Subproblem 2
r + 18.25 = -25.558217856 Simplifying r + 18.25 = -25.558217856 Reorder the terms: 18.25 + r = -25.558217856 Solving 18.25 + r = -25.558217856 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-18.25' to each side of the equation. 18.25 + -18.25 + r = -25.558217856 + -18.25 Combine like terms: 18.25 + -18.25 = 0.00 0.00 + r = -25.558217856 + -18.25 r = -25.558217856 + -18.25 Combine like terms: -25.558217856 + -18.25 = -43.808217856 r = -43.808217856 Simplifying r = -43.808217856Solution
The solution to the problem is based on the solutions from the subproblems. r = {7.308217856, -43.808217856}
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