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Simplifying 141.5512 = 6.28r2 + 6.28r(7.50) Reorder the terms for easier multiplication: 141.5512 = 6.28r2 + 6.28 * 7.50r Multiply 6.28 * 7.50 141.5512 = 6.28r2 + 47.1r Reorder the terms: 141.5512 = 47.1r + 6.28r2 Solving 141.5512 = 47.1r + 6.28r2 Solving for variable 'r'. Reorder the terms: 141.5512 + -47.1r + -6.28r2 = 47.1r + -47.1r + 6.28r2 + -6.28r2 Combine like terms: 47.1r + -47.1r = 0.0 141.5512 + -47.1r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2 141.5512 + -47.1r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 141.5512 + -47.1r + -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'. -22.54 + 7.5r + r2 = 0 Move the constant term to the right: Add '22.54' to each side of the equation. -22.54 + 7.5r + 22.54 + r2 = 0 + 22.54 Reorder the terms: -22.54 + 22.54 + 7.5r + r2 = 0 + 22.54 Combine like terms: -22.54 + 22.54 = 0.00 0.00 + 7.5r + r2 = 0 + 22.54 7.5r + r2 = 0 + 22.54 Combine like terms: 0 + 22.54 = 22.54 7.5r + r2 = 22.54 The r term is 7.5r. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5r + 14.0625 + r2 = 22.54 + 14.0625 Reorder the terms: 14.0625 + 7.5r + r2 = 22.54 + 14.0625 Combine like terms: 22.54 + 14.0625 = 36.6025 14.0625 + 7.5r + r2 = 36.6025 Factor a perfect square on the left side: (r + 3.75)(r + 3.75) = 36.6025 Calculate the square root of the right side: 6.05 Break this problem into two subproblems by setting (r + 3.75) equal to 6.05 and -6.05.Subproblem 1
r + 3.75 = 6.05 Simplifying r + 3.75 = 6.05 Reorder the terms: 3.75 + r = 6.05 Solving 3.75 + r = 6.05 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = 6.05 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = 6.05 + -3.75 r = 6.05 + -3.75 Combine like terms: 6.05 + -3.75 = 2.3 r = 2.3 Simplifying r = 2.3Subproblem 2
r + 3.75 = -6.05 Simplifying r + 3.75 = -6.05 Reorder the terms: 3.75 + r = -6.05 Solving 3.75 + r = -6.05 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + r = -6.05 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + r = -6.05 + -3.75 r = -6.05 + -3.75 Combine like terms: -6.05 + -3.75 = -9.8 r = -9.8 Simplifying r = -9.8Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.3, -9.8}
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