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Simplifying 2 * 3.14r2 + 2 * 3.14r(7.7) = 193.0472 Multiply 2 * 3.14 6.28r2 + 2 * 3.14r(7.7) = 193.0472 Reorder the terms for easier multiplication: 6.28r2 + 2 * 3.14 * 7.7r = 193.0472 Multiply 2 * 3.14 6.28r2 + 6.28 * 7.7r = 193.0472 Multiply 6.28 * 7.7 6.28r2 + 48.356r = 193.0472 Reorder the terms: 48.356r + 6.28r2 = 193.0472 Solving 48.356r + 6.28r2 = 193.0472 Solving for variable 'r'. Reorder the terms: -193.0472 + 48.356r + 6.28r2 = 193.0472 + -193.0472 Combine like terms: 193.0472 + -193.0472 = 0.0000 -193.0472 + 48.356r + 6.28r2 = 0.0000 Begin completing the square. Divide all terms by 6.28 the coefficient of the squared term: Divide each side by '6.28'. -30.74 + 7.7r + r2 = 0.00 Move the constant term to the right: Add '30.74' to each side of the equation. -30.74 + 7.7r + 30.74 + r2 = 0.00 + 30.74 Reorder the terms: -30.74 + 30.74 + 7.7r + r2 = 0.00 + 30.74 Combine like terms: -30.74 + 30.74 = 0.00 0.00 + 7.7r + r2 = 0.00 + 30.74 7.7r + r2 = 0.00 + 30.74 Combine like terms: 0.00 + 30.74 = 30.74 7.7r + r2 = 30.74 The r term is 7.7r. Take half its coefficient (3.85). Square it (14.8225) and add it to both sides. Add '14.8225' to each side of the equation. 7.7r + 14.8225 + r2 = 30.74 + 14.8225 Reorder the terms: 14.8225 + 7.7r + r2 = 30.74 + 14.8225 Combine like terms: 30.74 + 14.8225 = 45.5625 14.8225 + 7.7r + r2 = 45.5625 Factor a perfect square on the left side: (r + 3.85)(r + 3.85) = 45.5625 Calculate the square root of the right side: 6.75 Break this problem into two subproblems by setting (r + 3.85) equal to 6.75 and -6.75.Subproblem 1
r + 3.85 = 6.75 Simplifying r + 3.85 = 6.75 Reorder the terms: 3.85 + r = 6.75 Solving 3.85 + r = 6.75 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.85' to each side of the equation. 3.85 + -3.85 + r = 6.75 + -3.85 Combine like terms: 3.85 + -3.85 = 0.00 0.00 + r = 6.75 + -3.85 r = 6.75 + -3.85 Combine like terms: 6.75 + -3.85 = 2.9 r = 2.9 Simplifying r = 2.9Subproblem 2
r + 3.85 = -6.75 Simplifying r + 3.85 = -6.75 Reorder the terms: 3.85 + r = -6.75 Solving 3.85 + r = -6.75 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3.85' to each side of the equation. 3.85 + -3.85 + r = -6.75 + -3.85 Combine like terms: 3.85 + -3.85 = 0.00 0.00 + r = -6.75 + -3.85 r = -6.75 + -3.85 Combine like terms: -6.75 + -3.85 = -10.6 r = -10.6 Simplifying r = -10.6Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.9, -10.6}
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