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Simplifying 0 = 6.28r2 + 629.256r + -3234.1372 Reorder the terms: 0 = -3234.1372 + 629.256r + 6.28r2 Solving 0 = -3234.1372 + 629.256r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 3234.1372 = 3234.1372 3234.1372 + -629.256r + -6.28r2 = -3234.1372 + 629.256r + 6.28r2 + 3234.1372 + -629.256r + -6.28r2 Reorder the terms: 3234.1372 + -629.256r + -6.28r2 = -3234.1372 + 3234.1372 + 629.256r + -629.256r + 6.28r2 + -6.28r2 Combine like terms: -3234.1372 + 3234.1372 = 0.0000 3234.1372 + -629.256r + -6.28r2 = 0.0000 + 629.256r + -629.256r + 6.28r2 + -6.28r2 3234.1372 + -629.256r + -6.28r2 = 629.256r + -629.256r + 6.28r2 + -6.28r2 Combine like terms: 629.256r + -629.256r = 0.000 3234.1372 + -629.256r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 3234.1372 + -629.256r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 3234.1372 + -629.256r + -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'. -514.99 + 100.2r + r2 = 0 Move the constant term to the right: Add '514.99' to each side of the equation. -514.99 + 100.2r + 514.99 + r2 = 0 + 514.99 Reorder the terms: -514.99 + 514.99 + 100.2r + r2 = 0 + 514.99 Combine like terms: -514.99 + 514.99 = 0.00 0.00 + 100.2r + r2 = 0 + 514.99 100.2r + r2 = 0 + 514.99 Combine like terms: 0 + 514.99 = 514.99 100.2r + r2 = 514.99 The r term is 100.2r. Take half its coefficient (50.1). Square it (2510.01) and add it to both sides. Add '2510.01' to each side of the equation. 100.2r + 2510.01 + r2 = 514.99 + 2510.01 Reorder the terms: 2510.01 + 100.2r + r2 = 514.99 + 2510.01 Combine like terms: 514.99 + 2510.01 = 3025 2510.01 + 100.2r + r2 = 3025 Factor a perfect square on the left side: (r + 50.1)(r + 50.1) = 3025 Calculate the square root of the right side: 55 Break this problem into two subproblems by setting (r + 50.1) equal to 55 and -55.Subproblem 1
r + 50.1 = 55 Simplifying r + 50.1 = 55 Reorder the terms: 50.1 + r = 55 Solving 50.1 + r = 55 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-50.1' to each side of the equation. 50.1 + -50.1 + r = 55 + -50.1 Combine like terms: 50.1 + -50.1 = 0.0 0.0 + r = 55 + -50.1 r = 55 + -50.1 Combine like terms: 55 + -50.1 = 4.9 r = 4.9 Simplifying r = 4.9Subproblem 2
r + 50.1 = -55 Simplifying r + 50.1 = -55 Reorder the terms: 50.1 + r = -55 Solving 50.1 + r = -55 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-50.1' to each side of the equation. 50.1 + -50.1 + r = -55 + -50.1 Combine like terms: 50.1 + -50.1 = 0.0 0.0 + r = -55 + -50.1 r = -55 + -50.1 Combine like terms: -55 + -50.1 = -105.1 r = -105.1 Simplifying r = -105.1Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.9, -105.1}
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