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Simplifying 2 * 3.14r2 + 2 * 3.14r * 5 = 150.72 Multiply 2 * 3.14 6.28r2 + 2 * 3.14r * 5 = 150.72 Reorder the terms for easier multiplication: 6.28r2 + 2 * 3.14 * 5r = 150.72 Multiply 2 * 3.14 6.28r2 + 6.28 * 5r = 150.72 Multiply 6.28 * 5 6.28r2 + 31.4r = 150.72 Reorder the terms: 31.4r + 6.28r2 = 150.72 Solving 31.4r + 6.28r2 = 150.72 Solving for variable 'r'. Reorder the terms: -150.72 + 31.4r + 6.28r2 = 150.72 + -150.72 Combine like terms: 150.72 + -150.72 = 0.00 -150.72 + 31.4r + 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'. -24 + 5r + r2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 5r + 24 + r2 = 0 + 24 Reorder the terms: -24 + 24 + 5r + r2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 5r + r2 = 0 + 24 5r + r2 = 0 + 24 Combine like terms: 0 + 24 = 24 5r + r2 = 24 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 = 24 + 6.25 Reorder the terms: 6.25 + 5r + r2 = 24 + 6.25 Combine like terms: 24 + 6.25 = 30.25 6.25 + 5r + r2 = 30.25 Factor a perfect square on the left side: (r + 2.5)(r + 2.5) = 30.25 Calculate the square root of the right side: 5.5 Break this problem into two subproblems by setting (r + 2.5) equal to 5.5 and -5.5.Subproblem 1
r + 2.5 = 5.5 Simplifying r + 2.5 = 5.5 Reorder the terms: 2.5 + r = 5.5 Solving 2.5 + r = 5.5 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 = 5.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 5.5 + -2.5 r = 5.5 + -2.5 Combine like terms: 5.5 + -2.5 = 3 r = 3 Simplifying r = 3Subproblem 2
r + 2.5 = -5.5 Simplifying r + 2.5 = -5.5 Reorder the terms: 2.5 + r = -5.5 Solving 2.5 + r = -5.5 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 = -5.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -5.5 + -2.5 r = -5.5 + -2.5 Combine like terms: -5.5 + -2.5 = -8 r = -8 Simplifying r = -8Solution
The solution to the problem is based on the solutions from the subproblems. r = {3, -8}
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