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Simplifying 4148.9448 = 2 * 3.14r2 + 2 * 3.14r + 42.5 Multiply 2 * 3.14 4148.9448 = 6.28r2 + 2 * 3.14r + 42.5 Multiply 2 * 3.14 4148.9448 = 6.28r2 + 6.28r + 42.5 Reorder the terms: 4148.9448 = 42.5 + 6.28r + 6.28r2 Solving 4148.9448 = 42.5 + 6.28r + 6.28r2 Solving for variable 'r'. Combine like terms: 4148.9448 + -42.5 = 4106.4448 4106.4448 + -6.28r + -6.28r2 = 42.5 + 6.28r + 6.28r2 + -42.5 + -6.28r + -6.28r2 Reorder the terms: 4106.4448 + -6.28r + -6.28r2 = 42.5 + -42.5 + 6.28r + -6.28r + 6.28r2 + -6.28r2 Combine like terms: 42.5 + -42.5 = 0.0 4106.4448 + -6.28r + -6.28r2 = 0.0 + 6.28r + -6.28r + 6.28r2 + -6.28r2 4106.4448 + -6.28r + -6.28r2 = 6.28r + -6.28r + 6.28r2 + -6.28r2 Combine like terms: 6.28r + -6.28r = 0.00 4106.4448 + -6.28r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 4106.4448 + -6.28r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 4106.4448 + -6.28r + -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'. -653.8924841 + r + r2 = 0 Move the constant term to the right: Add '653.8924841' to each side of the equation. -653.8924841 + r + 653.8924841 + r2 = 0 + 653.8924841 Reorder the terms: -653.8924841 + 653.8924841 + r + r2 = 0 + 653.8924841 Combine like terms: -653.8924841 + 653.8924841 = 0.0000000 0.0000000 + r + r2 = 0 + 653.8924841 r + r2 = 0 + 653.8924841 Combine like terms: 0 + 653.8924841 = 653.8924841 r + r2 = 653.8924841 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 653.8924841 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 653.8924841 + 0.25 Combine like terms: 653.8924841 + 0.25 = 654.1424841 1.25 + r2 = 654.1424841 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 654.1424841 Calculate the square root of the right side: 25.576209338 Break this problem into two subproblems by setting (r + 0.5) equal to 25.576209338 and -25.576209338.Subproblem 1
r + 0.5 = 25.576209338 Simplifying r + 0.5 = 25.576209338 Reorder the terms: 0.5 + r = 25.576209338 Solving 0.5 + r = 25.576209338 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 25.576209338 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 25.576209338 + -0.5 r = 25.576209338 + -0.5 Combine like terms: 25.576209338 + -0.5 = 25.076209338 r = 25.076209338 Simplifying r = 25.076209338Subproblem 2
r + 0.5 = -25.576209338 Simplifying r + 0.5 = -25.576209338 Reorder the terms: 0.5 + r = -25.576209338 Solving 0.5 + r = -25.576209338 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -25.576209338 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -25.576209338 + -0.5 r = -25.576209338 + -0.5 Combine like terms: -25.576209338 + -0.5 = -26.076209338 r = -26.076209338 Simplifying r = -26.076209338Solution
The solution to the problem is based on the solutions from the subproblems. r = {25.076209338, -26.076209338}
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