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Simplifying 633.024 = 6.28r2 + 101.736r Reorder the terms: 633.024 = 101.736r + 6.28r2 Solving 633.024 = 101.736r + 6.28r2 Solving for variable 'r'. Reorder the terms: 633.024 + -101.736r + -6.28r2 = 101.736r + -101.736r + 6.28r2 + -6.28r2 Combine like terms: 101.736r + -101.736r = 0.000 633.024 + -101.736r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 633.024 + -101.736r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 633.024 + -101.736r + -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'. -100.8 + 16.2r + r2 = 0 Move the constant term to the right: Add '100.8' to each side of the equation. -100.8 + 16.2r + 100.8 + r2 = 0 + 100.8 Reorder the terms: -100.8 + 100.8 + 16.2r + r2 = 0 + 100.8 Combine like terms: -100.8 + 100.8 = 0.0 0.0 + 16.2r + r2 = 0 + 100.8 16.2r + r2 = 0 + 100.8 Combine like terms: 0 + 100.8 = 100.8 16.2r + r2 = 100.8 The r term is 16.2r. Take half its coefficient (8.1). Square it (65.61) and add it to both sides. Add '65.61' to each side of the equation. 16.2r + 65.61 + r2 = 100.8 + 65.61 Reorder the terms: 65.61 + 16.2r + r2 = 100.8 + 65.61 Combine like terms: 100.8 + 65.61 = 166.41 65.61 + 16.2r + r2 = 166.41 Factor a perfect square on the left side: (r + 8.1)(r + 8.1) = 166.41 Calculate the square root of the right side: 12.9 Break this problem into two subproblems by setting (r + 8.1) equal to 12.9 and -12.9.Subproblem 1
r + 8.1 = 12.9 Simplifying r + 8.1 = 12.9 Reorder the terms: 8.1 + r = 12.9 Solving 8.1 + r = 12.9 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = 12.9 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = 12.9 + -8.1 r = 12.9 + -8.1 Combine like terms: 12.9 + -8.1 = 4.8 r = 4.8 Simplifying r = 4.8Subproblem 2
r + 8.1 = -12.9 Simplifying r + 8.1 = -12.9 Reorder the terms: 8.1 + r = -12.9 Solving 8.1 + r = -12.9 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = -12.9 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = -12.9 + -8.1 r = -12.9 + -8.1 Combine like terms: -12.9 + -8.1 = -21 r = -21 Simplifying r = -21Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.8, -21}
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