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Simplifying 4148.9448 = 6.28r2 + (6.28) * r(42.5) Reorder the terms for easier multiplication: 4148.9448 = 6.28r2 + 6.28 * 42.5r Multiply 6.28 * 42.5 4148.9448 = 6.28r2 + 266.9r Reorder the terms: 4148.9448 = 266.9r + 6.28r2 Solving 4148.9448 = 266.9r + 6.28r2 Solving for variable 'r'. Reorder the terms: 4148.9448 + -266.9r + -6.28r2 = 266.9r + -266.9r + 6.28r2 + -6.28r2 Combine like terms: 266.9r + -266.9r = 0.0 4148.9448 + -266.9r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 4148.9448 + -266.9r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 4148.9448 + -266.9r + -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'. -660.66 + 42.5r + r2 = 0 Move the constant term to the right: Add '660.66' to each side of the equation. -660.66 + 42.5r + 660.66 + r2 = 0 + 660.66 Reorder the terms: -660.66 + 660.66 + 42.5r + r2 = 0 + 660.66 Combine like terms: -660.66 + 660.66 = 0.00 0.00 + 42.5r + r2 = 0 + 660.66 42.5r + r2 = 0 + 660.66 Combine like terms: 0 + 660.66 = 660.66 42.5r + r2 = 660.66 The r term is 42.5r. Take half its coefficient (21.25). Square it (451.5625) and add it to both sides. Add '451.5625' to each side of the equation. 42.5r + 451.5625 + r2 = 660.66 + 451.5625 Reorder the terms: 451.5625 + 42.5r + r2 = 660.66 + 451.5625 Combine like terms: 660.66 + 451.5625 = 1112.2225 451.5625 + 42.5r + r2 = 1112.2225 Factor a perfect square on the left side: (r + 21.25)(r + 21.25) = 1112.2225 Calculate the square root of the right side: 33.35 Break this problem into two subproblems by setting (r + 21.25) equal to 33.35 and -33.35.Subproblem 1
r + 21.25 = 33.35 Simplifying r + 21.25 = 33.35 Reorder the terms: 21.25 + r = 33.35 Solving 21.25 + r = 33.35 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-21.25' to each side of the equation. 21.25 + -21.25 + r = 33.35 + -21.25 Combine like terms: 21.25 + -21.25 = 0.00 0.00 + r = 33.35 + -21.25 r = 33.35 + -21.25 Combine like terms: 33.35 + -21.25 = 12.1 r = 12.1 Simplifying r = 12.1Subproblem 2
r + 21.25 = -33.35 Simplifying r + 21.25 = -33.35 Reorder the terms: 21.25 + r = -33.35 Solving 21.25 + r = -33.35 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-21.25' to each side of the equation. 21.25 + -21.25 + r = -33.35 + -21.25 Combine like terms: 21.25 + -21.25 = 0.00 0.00 + r = -33.35 + -21.25 r = -33.35 + -21.25 Combine like terms: -33.35 + -21.25 = -54.6 r = -54.6 Simplifying r = -54.6Solution
The solution to the problem is based on the solutions from the subproblems. r = {12.1, -54.6}
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