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Simplifying 355.134 = 6.28r2 + 6.28r * 10.6 Reorder the terms for easier multiplication: 355.134 = 6.28r2 + 6.28 * 10.6r Multiply 6.28 * 10.6 355.134 = 6.28r2 + 66.568r Reorder the terms: 355.134 = 66.568r + 6.28r2 Solving 355.134 = 66.568r + 6.28r2 Solving for variable 'r'. Reorder the terms: 355.134 + -66.568r + -6.28r2 = 66.568r + -66.568r + 6.28r2 + -6.28r2 Combine like terms: 66.568r + -66.568r = 0.000 355.134 + -66.568r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 355.134 + -66.568r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 355.134 + -66.568r + -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'. -56.55 + 10.6r + r2 = 0 Move the constant term to the right: Add '56.55' to each side of the equation. -56.55 + 10.6r + 56.55 + r2 = 0 + 56.55 Reorder the terms: -56.55 + 56.55 + 10.6r + r2 = 0 + 56.55 Combine like terms: -56.55 + 56.55 = 0.00 0.00 + 10.6r + r2 = 0 + 56.55 10.6r + r2 = 0 + 56.55 Combine like terms: 0 + 56.55 = 56.55 10.6r + r2 = 56.55 The r term is 10.6r. Take half its coefficient (5.3). Square it (28.09) and add it to both sides. Add '28.09' to each side of the equation. 10.6r + 28.09 + r2 = 56.55 + 28.09 Reorder the terms: 28.09 + 10.6r + r2 = 56.55 + 28.09 Combine like terms: 56.55 + 28.09 = 84.64 28.09 + 10.6r + r2 = 84.64 Factor a perfect square on the left side: (r + 5.3)(r + 5.3) = 84.64 Calculate the square root of the right side: 9.2 Break this problem into two subproblems by setting (r + 5.3) equal to 9.2 and -9.2.Subproblem 1
r + 5.3 = 9.2 Simplifying r + 5.3 = 9.2 Reorder the terms: 5.3 + r = 9.2 Solving 5.3 + r = 9.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5.3' to each side of the equation. 5.3 + -5.3 + r = 9.2 + -5.3 Combine like terms: 5.3 + -5.3 = 0.0 0.0 + r = 9.2 + -5.3 r = 9.2 + -5.3 Combine like terms: 9.2 + -5.3 = 3.9 r = 3.9 Simplifying r = 3.9Subproblem 2
r + 5.3 = -9.2 Simplifying r + 5.3 = -9.2 Reorder the terms: 5.3 + r = -9.2 Solving 5.3 + r = -9.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5.3' to each side of the equation. 5.3 + -5.3 + r = -9.2 + -5.3 Combine like terms: 5.3 + -5.3 = 0.0 0.0 + r = -9.2 + -5.3 r = -9.2 + -5.3 Combine like terms: -9.2 + -5.3 = -14.5 r = -14.5 Simplifying r = -14.5Solution
The solution to the problem is based on the solutions from the subproblems. r = {3.9, -14.5}
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