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Simplifying 2 * 3.14r2 + 2 * 3.14r * 5.3 = 285.9918 Multiply 2 * 3.14 6.28r2 + 2 * 3.14r * 5.3 = 285.9918 Reorder the terms for easier multiplication: 6.28r2 + 2 * 3.14 * 5.3r = 285.9918 Multiply 2 * 3.14 6.28r2 + 6.28 * 5.3r = 285.9918 Multiply 6.28 * 5.3 6.28r2 + 33.284r = 285.9918 Reorder the terms: 33.284r + 6.28r2 = 285.9918 Solving 33.284r + 6.28r2 = 285.9918 Solving for variable 'r'. Reorder the terms: -285.9918 + 33.284r + 6.28r2 = 285.9918 + -285.9918 Combine like terms: 285.9918 + -285.9918 = 0.0000 -285.9918 + 33.284r + 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'. -45.54009554 + 5.3r + r2 = 0.00 Move the constant term to the right: Add '45.54009554' to each side of the equation. -45.54009554 + 5.3r + 45.54009554 + r2 = 0.00 + 45.54009554 Reorder the terms: -45.54009554 + 45.54009554 + 5.3r + r2 = 0.00 + 45.54009554 Combine like terms: -45.54009554 + 45.54009554 = 0.00000000 0.00000000 + 5.3r + r2 = 0.00 + 45.54009554 5.3r + r2 = 0.00 + 45.54009554 Combine like terms: 0.00 + 45.54009554 = 45.54009554 5.3r + r2 = 45.54009554 The r term is 5.3r. Take half its coefficient (2.65). Square it (7.0225) and add it to both sides. Add '7.0225' to each side of the equation. 5.3r + 7.0225 + r2 = 45.54009554 + 7.0225 Reorder the terms: 7.0225 + 5.3r + r2 = 45.54009554 + 7.0225 Combine like terms: 45.54009554 + 7.0225 = 52.56259554 7.0225 + 5.3r + r2 = 52.56259554 Factor a perfect square on the left side: (r + 2.65)(r + 2.65) = 52.56259554 Calculate the square root of the right side: 7.250006589 Break this problem into two subproblems by setting (r + 2.65) equal to 7.250006589 and -7.250006589.Subproblem 1
r + 2.65 = 7.250006589 Simplifying r + 2.65 = 7.250006589 Reorder the terms: 2.65 + r = 7.250006589 Solving 2.65 + r = 7.250006589 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = 7.250006589 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = 7.250006589 + -2.65 r = 7.250006589 + -2.65 Combine like terms: 7.250006589 + -2.65 = 4.600006589 r = 4.600006589 Simplifying r = 4.600006589Subproblem 2
r + 2.65 = -7.250006589 Simplifying r + 2.65 = -7.250006589 Reorder the terms: 2.65 + r = -7.250006589 Solving 2.65 + r = -7.250006589 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = -7.250006589 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = -7.250006589 + -2.65 r = -7.250006589 + -2.65 Combine like terms: -7.250006589 + -2.65 = -9.900006589 r = -9.900006589 Simplifying r = -9.900006589Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.600006589, -9.900006589}
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