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Simplifying 0 = 6.28r2 + 53.38r + -181.2408 Reorder the terms: 0 = -181.2408 + 53.38r + 6.28r2 Solving 0 = -181.2408 + 53.38r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 181.2408 = 181.2408 181.2408 + -53.38r + -6.28r2 = -181.2408 + 53.38r + 6.28r2 + 181.2408 + -53.38r + -6.28r2 Reorder the terms: 181.2408 + -53.38r + -6.28r2 = -181.2408 + 181.2408 + 53.38r + -53.38r + 6.28r2 + -6.28r2 Combine like terms: -181.2408 + 181.2408 = 0.0000 181.2408 + -53.38r + -6.28r2 = 0.0000 + 53.38r + -53.38r + 6.28r2 + -6.28r2 181.2408 + -53.38r + -6.28r2 = 53.38r + -53.38r + 6.28r2 + -6.28r2 Combine like terms: 53.38r + -53.38r = 0.00 181.2408 + -53.38r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 181.2408 + -53.38r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 181.2408 + -53.38r + -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'. -28.86 + 8.5r + r2 = 0 Move the constant term to the right: Add '28.86' to each side of the equation. -28.86 + 8.5r + 28.86 + r2 = 0 + 28.86 Reorder the terms: -28.86 + 28.86 + 8.5r + r2 = 0 + 28.86 Combine like terms: -28.86 + 28.86 = 0.00 0.00 + 8.5r + r2 = 0 + 28.86 8.5r + r2 = 0 + 28.86 Combine like terms: 0 + 28.86 = 28.86 8.5r + r2 = 28.86 The r term is 8.5r. Take half its coefficient (4.25). Square it (18.0625) and add it to both sides. Add '18.0625' to each side of the equation. 8.5r + 18.0625 + r2 = 28.86 + 18.0625 Reorder the terms: 18.0625 + 8.5r + r2 = 28.86 + 18.0625 Combine like terms: 28.86 + 18.0625 = 46.9225 18.0625 + 8.5r + r2 = 46.9225 Factor a perfect square on the left side: (r + 4.25)(r + 4.25) = 46.9225 Calculate the square root of the right side: 6.85 Break this problem into two subproblems by setting (r + 4.25) equal to 6.85 and -6.85.Subproblem 1
r + 4.25 = 6.85 Simplifying r + 4.25 = 6.85 Reorder the terms: 4.25 + r = 6.85 Solving 4.25 + r = 6.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + r = 6.85 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + r = 6.85 + -4.25 r = 6.85 + -4.25 Combine like terms: 6.85 + -4.25 = 2.6 r = 2.6 Simplifying r = 2.6Subproblem 2
r + 4.25 = -6.85 Simplifying r + 4.25 = -6.85 Reorder the terms: 4.25 + r = -6.85 Solving 4.25 + r = -6.85 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + r = -6.85 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + r = -6.85 + -4.25 r = -6.85 + -4.25 Combine like terms: -6.85 + -4.25 = -11.1 r = -11.1 Simplifying r = -11.1Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.6, -11.1}
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