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Simplifying 0 = 6.28r2 + 81.64r + -453.6672 Reorder the terms: 0 = -453.6672 + 81.64r + 6.28r2 Solving 0 = -453.6672 + 81.64r + 6.28r2 Solving for variable 'r'. Combine like terms: 0 + 453.6672 = 453.6672 453.6672 + -81.64r + -6.28r2 = -453.6672 + 81.64r + 6.28r2 + 453.6672 + -81.64r + -6.28r2 Reorder the terms: 453.6672 + -81.64r + -6.28r2 = -453.6672 + 453.6672 + 81.64r + -81.64r + 6.28r2 + -6.28r2 Combine like terms: -453.6672 + 453.6672 = 0.0000 453.6672 + -81.64r + -6.28r2 = 0.0000 + 81.64r + -81.64r + 6.28r2 + -6.28r2 453.6672 + -81.64r + -6.28r2 = 81.64r + -81.64r + 6.28r2 + -6.28r2 Combine like terms: 81.64r + -81.64r = 0.00 453.6672 + -81.64r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2 453.6672 + -81.64r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 453.6672 + -81.64r + -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'. -72.24 + 13r + r2 = 0 Move the constant term to the right: Add '72.24' to each side of the equation. -72.24 + 13r + 72.24 + r2 = 0 + 72.24 Reorder the terms: -72.24 + 72.24 + 13r + r2 = 0 + 72.24 Combine like terms: -72.24 + 72.24 = 0.00 0.00 + 13r + r2 = 0 + 72.24 13r + r2 = 0 + 72.24 Combine like terms: 0 + 72.24 = 72.24 13r + r2 = 72.24 The r term is 13r. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13r + 42.25 + r2 = 72.24 + 42.25 Reorder the terms: 42.25 + 13r + r2 = 72.24 + 42.25 Combine like terms: 72.24 + 42.25 = 114.49 42.25 + 13r + r2 = 114.49 Factor a perfect square on the left side: (r + 6.5)(r + 6.5) = 114.49 Calculate the square root of the right side: 10.7 Break this problem into two subproblems by setting (r + 6.5) equal to 10.7 and -10.7.Subproblem 1
r + 6.5 = 10.7 Simplifying r + 6.5 = 10.7 Reorder the terms: 6.5 + r = 10.7 Solving 6.5 + r = 10.7 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + r = 10.7 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + r = 10.7 + -6.5 r = 10.7 + -6.5 Combine like terms: 10.7 + -6.5 = 4.2 r = 4.2 Simplifying r = 4.2Subproblem 2
r + 6.5 = -10.7 Simplifying r + 6.5 = -10.7 Reorder the terms: 6.5 + r = -10.7 Solving 6.5 + r = -10.7 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + r = -10.7 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + r = -10.7 + -6.5 r = -10.7 + -6.5 Combine like terms: -10.7 + -6.5 = -17.2 r = -17.2 Simplifying r = -17.2Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.2, -17.2}
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