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Simplifying 285.9912 = 6.28r2 + 6.28r * 5.3 Reorder the terms for easier multiplication: 285.9912 = 6.28r2 + 6.28 * 5.3r Multiply 6.28 * 5.3 285.9912 = 6.28r2 + 33.284r Reorder the terms: 285.9912 = 33.284r + 6.28r2 Solving 285.9912 = 33.284r + 6.28r2 Solving for variable 'r'. Reorder the terms: 285.9912 + -33.284r + -6.28r2 = 33.284r + -33.284r + 6.28r2 + -6.28r2 Combine like terms: 33.284r + -33.284r = 0.000 285.9912 + -33.284r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 285.9912 + -33.284r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 285.9912 + -33.284r + -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'. -45.54 + 5.3r + r2 = 0 Move the constant term to the right: Add '45.54' to each side of the equation. -45.54 + 5.3r + 45.54 + r2 = 0 + 45.54 Reorder the terms: -45.54 + 45.54 + 5.3r + r2 = 0 + 45.54 Combine like terms: -45.54 + 45.54 = 0.00 0.00 + 5.3r + r2 = 0 + 45.54 5.3r + r2 = 0 + 45.54 Combine like terms: 0 + 45.54 = 45.54 5.3r + r2 = 45.54 The r term is 5.3r. Take half its coefficient (2.65). Square it (7.0225) and add it to both sides. Add '7.0225' to each side of the equation. 5.3r + 7.0225 + r2 = 45.54 + 7.0225 Reorder the terms: 7.0225 + 5.3r + r2 = 45.54 + 7.0225 Combine like terms: 45.54 + 7.0225 = 52.5625 7.0225 + 5.3r + r2 = 52.5625 Factor a perfect square on the left side: (r + 2.65)(r + 2.65) = 52.5625 Calculate the square root of the right side: 7.25 Break this problem into two subproblems by setting (r + 2.65) equal to 7.25 and -7.25.Subproblem 1
r + 2.65 = 7.25 Simplifying r + 2.65 = 7.25 Reorder the terms: 2.65 + r = 7.25 Solving 2.65 + r = 7.25 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = 7.25 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = 7.25 + -2.65 r = 7.25 + -2.65 Combine like terms: 7.25 + -2.65 = 4.6 r = 4.6 Simplifying r = 4.6Subproblem 2
r + 2.65 = -7.25 Simplifying r + 2.65 = -7.25 Reorder the terms: 2.65 + r = -7.25 Solving 2.65 + r = -7.25 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = -7.25 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = -7.25 + -2.65 r = -7.25 + -2.65 Combine like terms: -7.25 + -2.65 = -9.9 r = -9.9 Simplifying r = -9.9Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.6, -9.9}
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