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Simplifying 0 = 6.28r2 + 390.616r + -5645.9084 Reorder the terms: 0 = -5645.9084 + 390.616r + 6.28r2 Solving 0 = -5645.9084 + 390.616r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 5645.9084 = 5645.9084 5645.9084 + -390.616r + -6.28r2 = -5645.9084 + 390.616r + 6.28r2 + 5645.9084 + -390.616r + -6.28r2 Reorder the terms: 5645.9084 + -390.616r + -6.28r2 = -5645.9084 + 5645.9084 + 390.616r + -390.616r + 6.28r2 + -6.28r2 Combine like terms: -5645.9084 + 5645.9084 = 0.0000 5645.9084 + -390.616r + -6.28r2 = 0.0000 + 390.616r + -390.616r + 6.28r2 + -6.28r2 5645.9084 + -390.616r + -6.28r2 = 390.616r + -390.616r + 6.28r2 + -6.28r2 Combine like terms: 390.616r + -390.616r = 0.000 5645.9084 + -390.616r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 5645.9084 + -390.616r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 5645.9084 + -390.616r + -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'. -899.03 + 62.2r + r2 = 0 Move the constant term to the right: Add '899.03' to each side of the equation. -899.03 + 62.2r + 899.03 + r2 = 0 + 899.03 Reorder the terms: -899.03 + 899.03 + 62.2r + r2 = 0 + 899.03 Combine like terms: -899.03 + 899.03 = 0.00 0.00 + 62.2r + r2 = 0 + 899.03 62.2r + r2 = 0 + 899.03 Combine like terms: 0 + 899.03 = 899.03 62.2r + r2 = 899.03 The r term is 62.2r. Take half its coefficient (31.1). Square it (967.21) and add it to both sides. Add '967.21' to each side of the equation. 62.2r + 967.21 + r2 = 899.03 + 967.21 Reorder the terms: 967.21 + 62.2r + r2 = 899.03 + 967.21 Combine like terms: 899.03 + 967.21 = 1866.24 967.21 + 62.2r + r2 = 1866.24 Factor a perfect square on the left side: (r + 31.1)(r + 31.1) = 1866.24 Calculate the square root of the right side: 43.2 Break this problem into two subproblems by setting (r + 31.1) equal to 43.2 and -43.2.Subproblem 1
r + 31.1 = 43.2 Simplifying r + 31.1 = 43.2 Reorder the terms: 31.1 + r = 43.2 Solving 31.1 + r = 43.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-31.1' to each side of the equation. 31.1 + -31.1 + r = 43.2 + -31.1 Combine like terms: 31.1 + -31.1 = 0.0 0.0 + r = 43.2 + -31.1 r = 43.2 + -31.1 Combine like terms: 43.2 + -31.1 = 12.1 r = 12.1 Simplifying r = 12.1Subproblem 2
r + 31.1 = -43.2 Simplifying r + 31.1 = -43.2 Reorder the terms: 31.1 + r = -43.2 Solving 31.1 + r = -43.2 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-31.1' to each side of the equation. 31.1 + -31.1 + r = -43.2 + -31.1 Combine like terms: 31.1 + -31.1 = 0.0 0.0 + r = -43.2 + -31.1 r = -43.2 + -31.1 Combine like terms: -43.2 + -31.1 = -74.3 r = -74.3 Simplifying r = -74.3Solution
The solution to the problem is based on the solutions from the subproblems. r = {12.1, -74.3}
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