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Simplifying 2 * 3.14r * 5 + 2 * 3.14r2 = 48 Reorder the terms for easier multiplication: 2 * 3.14 * 5r + 2 * 3.14r2 = 48 Multiply 2 * 3.14 6.28 * 5r + 2 * 3.14r2 = 48 Multiply 6.28 * 5 31.4r + 2 * 3.14r2 = 48 Multiply 2 * 3.14 31.4r + 6.28r2 = 48 Solving 31.4r + 6.28r2 = 48 Solving for variable 'r'. Reorder the terms: -48 + 31.4r + 6.28r2 = 48 + -48 Combine like terms: 48 + -48 = 0 -48 + 31.4r + 6.28r2 = 0 Begin completing the square. Divide all terms by 6.28 the coefficient of the squared term: Divide each side by '6.28'. -7.643312102 + 5r + r2 = 0 Move the constant term to the right: Add '7.643312102' to each side of the equation. -7.643312102 + 5r + 7.643312102 + r2 = 0 + 7.643312102 Reorder the terms: -7.643312102 + 7.643312102 + 5r + r2 = 0 + 7.643312102 Combine like terms: -7.643312102 + 7.643312102 = 0.000000000 0.000000000 + 5r + r2 = 0 + 7.643312102 5r + r2 = 0 + 7.643312102 Combine like terms: 0 + 7.643312102 = 7.643312102 5r + r2 = 7.643312102 The r term is 5r. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5r + 6.25 + r2 = 7.643312102 + 6.25 Reorder the terms: 6.25 + 5r + r2 = 7.643312102 + 6.25 Combine like terms: 7.643312102 + 6.25 = 13.893312102 6.25 + 5r + r2 = 13.893312102 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 13.893312102 Calculate the square root of the right side: 3.727373352 Break this problem into two subproblems by setting (r + 2.5) equal to 3.727373352 and -3.727373352.Subproblem 1
r + 2.5 = 3.727373352 Simplifying r + 2.5 = 3.727373352 Reorder the terms: 2.5 + r = 3.727373352 Solving 2.5 + r = 3.727373352 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 3.727373352 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 3.727373352 + -2.5 r = 3.727373352 + -2.5 Combine like terms: 3.727373352 + -2.5 = 1.227373352 r = 1.227373352 Simplifying r = 1.227373352Subproblem 2
r + 2.5 = -3.727373352 Simplifying r + 2.5 = -3.727373352 Reorder the terms: 2.5 + r = -3.727373352 Solving 2.5 + r = -3.727373352 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -3.727373352 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -3.727373352 + -2.5 r = -3.727373352 + -2.5 Combine like terms: -3.727373352 + -2.5 = -6.227373352 r = -6.227373352 Simplifying r = -6.227373352Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.227373352, -6.227373352}
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