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Simplifying 569.7216 = 6.28r2 + 109.272r Reorder the terms: 569.7216 = 109.272r + 6.28r2 Solving 569.7216 = 109.272r + 6.28r2 Solving for variable 'r'. Reorder the terms: 569.7216 + -109.272r + -6.28r2 = 109.272r + -109.272r + 6.28r2 + -6.28r2 Combine like terms: 109.272r + -109.272r = 0.000 569.7216 + -109.272r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 569.7216 + -109.272r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 569.7216 + -109.272r + -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'. -90.72 + 17.4r + r2 = 0 Move the constant term to the right: Add '90.72' to each side of the equation. -90.72 + 17.4r + 90.72 + r2 = 0 + 90.72 Reorder the terms: -90.72 + 90.72 + 17.4r + r2 = 0 + 90.72 Combine like terms: -90.72 + 90.72 = 0.00 0.00 + 17.4r + r2 = 0 + 90.72 17.4r + r2 = 0 + 90.72 Combine like terms: 0 + 90.72 = 90.72 17.4r + r2 = 90.72 The r term is 17.4r. Take half its coefficient (8.7). Square it (75.69) and add it to both sides. Add '75.69' to each side of the equation. 17.4r + 75.69 + r2 = 90.72 + 75.69 Reorder the terms: 75.69 + 17.4r + r2 = 90.72 + 75.69 Combine like terms: 90.72 + 75.69 = 166.41 75.69 + 17.4r + r2 = 166.41 Factor a perfect square on the left side: (r + 8.7)(r + 8.7) = 166.41 Calculate the square root of the right side: 12.9 Break this problem into two subproblems by setting (r + 8.7) equal to 12.9 and -12.9.Subproblem 1
r + 8.7 = 12.9 Simplifying r + 8.7 = 12.9 Reorder the terms: 8.7 + r = 12.9 Solving 8.7 + r = 12.9 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.7' to each side of the equation. 8.7 + -8.7 + r = 12.9 + -8.7 Combine like terms: 8.7 + -8.7 = 0.0 0.0 + r = 12.9 + -8.7 r = 12.9 + -8.7 Combine like terms: 12.9 + -8.7 = 4.2 r = 4.2 Simplifying r = 4.2Subproblem 2
r + 8.7 = -12.9 Simplifying r + 8.7 = -12.9 Reorder the terms: 8.7 + r = -12.9 Solving 8.7 + r = -12.9 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.7' to each side of the equation. 8.7 + -8.7 + r = -12.9 + -8.7 Combine like terms: 8.7 + -8.7 = 0.0 0.0 + r = -12.9 + -8.7 r = -12.9 + -8.7 Combine like terms: -12.9 + -8.7 = -21.6 r = -21.6 Simplifying r = -21.6Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.2, -21.6}
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