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Simplifying 355.134 = 2 * 3.14r2 + 2.314r * 10.6 Multiply 2 * 3.14 355.134 = 6.28r2 + 2.314r * 10.6 Reorder the terms for easier multiplication: 355.134 = 6.28r2 + 2.314 * 10.6r Multiply 2.314 * 10.6 355.134 = 6.28r2 + 24.5284r Reorder the terms: 355.134 = 24.5284r + 6.28r2 Solving 355.134 = 24.5284r + 6.28r2 Solving for variable 'r'. Reorder the terms: 355.134 + -24.5284r + -6.28r2 = 24.5284r + -24.5284r + 6.28r2 + -6.28r2 Combine like terms: 24.5284r + -24.5284r = 0.0000 355.134 + -24.5284r + -6.28r2 = 0.0000 + 6.28r2 + -6.28r2 355.134 + -24.5284r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 355.134 + -24.5284r + -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'. -56.55 + 3.905796178r + r2 = 0 Move the constant term to the right: Add '56.55' to each side of the equation. -56.55 + 3.905796178r + 56.55 + r2 = 0 + 56.55 Reorder the terms: -56.55 + 56.55 + 3.905796178r + r2 = 0 + 56.55 Combine like terms: -56.55 + 56.55 = 0.00 0.00 + 3.905796178r + r2 = 0 + 56.55 3.905796178r + r2 = 0 + 56.55 Combine like terms: 0 + 56.55 = 56.55 3.905796178r + r2 = 56.55 The r term is 3.905796178r. Take half its coefficient (1.952898089). Square it (3.813810946) and add it to both sides. Add '3.813810946' to each side of the equation. 3.905796178r + 3.813810946 + r2 = 56.55 + 3.813810946 Reorder the terms: 3.813810946 + 3.905796178r + r2 = 56.55 + 3.813810946 Combine like terms: 56.55 + 3.813810946 = 60.363810946 3.813810946 + 3.905796178r + r2 = 60.363810946 Factor a perfect square on the left side: (r + 1.952898089)(r + 1.952898089) = 60.363810946 Calculate the square root of the right side: 7.769415097 Break this problem into two subproblems by setting (r + 1.952898089) equal to 7.769415097 and -7.769415097.Subproblem 1
r + 1.952898089 = 7.769415097 Simplifying r + 1.952898089 = 7.769415097 Reorder the terms: 1.952898089 + r = 7.769415097 Solving 1.952898089 + r = 7.769415097 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.952898089' to each side of the equation. 1.952898089 + -1.952898089 + r = 7.769415097 + -1.952898089 Combine like terms: 1.952898089 + -1.952898089 = 0.000000000 0.000000000 + r = 7.769415097 + -1.952898089 r = 7.769415097 + -1.952898089 Combine like terms: 7.769415097 + -1.952898089 = 5.816517008 r = 5.816517008 Simplifying r = 5.816517008Subproblem 2
r + 1.952898089 = -7.769415097 Simplifying r + 1.952898089 = -7.769415097 Reorder the terms: 1.952898089 + r = -7.769415097 Solving 1.952898089 + r = -7.769415097 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.952898089' to each side of the equation. 1.952898089 + -1.952898089 + r = -7.769415097 + -1.952898089 Combine like terms: 1.952898089 + -1.952898089 = 0.000000000 0.000000000 + r = -7.769415097 + -1.952898089 r = -7.769415097 + -1.952898089 Combine like terms: -7.769415097 + -1.952898089 = -9.722313186 r = -9.722313186 Simplifying r = -9.722313186Solution
The solution to the problem is based on the solutions from the subproblems. r = {5.816517008, -9.722313186}
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