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Simplifying 591.3876 = 6.28r2 + 110.528r Reorder the terms: 591.3876 = 110.528r + 6.28r2 Solving 591.3876 = 110.528r + 6.28r2 Solving for variable 'r'. Reorder the terms: 591.3876 + -110.528r + -6.28r2 = 110.528r + -110.528r + 6.28r2 + -6.28r2 Combine like terms: 110.528r + -110.528r = 0.000 591.3876 + -110.528r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 591.3876 + -110.528r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 591.3876 + -110.528r + -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'. -94.17 + 17.6r + r2 = 0 Move the constant term to the right: Add '94.17' to each side of the equation. -94.17 + 17.6r + 94.17 + r2 = 0 + 94.17 Reorder the terms: -94.17 + 94.17 + 17.6r + r2 = 0 + 94.17 Combine like terms: -94.17 + 94.17 = 0.00 0.00 + 17.6r + r2 = 0 + 94.17 17.6r + r2 = 0 + 94.17 Combine like terms: 0 + 94.17 = 94.17 17.6r + r2 = 94.17 The r term is 17.6r. Take half its coefficient (8.8). Square it (77.44) and add it to both sides. Add '77.44' to each side of the equation. 17.6r + 77.44 + r2 = 94.17 + 77.44 Reorder the terms: 77.44 + 17.6r + r2 = 94.17 + 77.44 Combine like terms: 94.17 + 77.44 = 171.61 77.44 + 17.6r + r2 = 171.61 Factor a perfect square on the left side: (r + 8.8)(r + 8.8) = 171.61 Calculate the square root of the right side: 13.1 Break this problem into two subproblems by setting (r + 8.8) equal to 13.1 and -13.1.Subproblem 1
r + 8.8 = 13.1 Simplifying r + 8.8 = 13.1 Reorder the terms: 8.8 + r = 13.1 Solving 8.8 + r = 13.1 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.8' to each side of the equation. 8.8 + -8.8 + r = 13.1 + -8.8 Combine like terms: 8.8 + -8.8 = 0.0 0.0 + r = 13.1 + -8.8 r = 13.1 + -8.8 Combine like terms: 13.1 + -8.8 = 4.3 r = 4.3 Simplifying r = 4.3Subproblem 2
r + 8.8 = -13.1 Simplifying r + 8.8 = -13.1 Reorder the terms: 8.8 + r = -13.1 Solving 8.8 + r = -13.1 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.8' to each side of the equation. 8.8 + -8.8 + r = -13.1 + -8.8 Combine like terms: 8.8 + -8.8 = 0.0 0.0 + r = -13.1 + -8.8 r = -13.1 + -8.8 Combine like terms: -13.1 + -8.8 = -21.9 r = -21.9 Simplifying r = -21.9Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.3, -21.9}
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