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Simplifying 189.028 = 6.28r2 + 16.956r Reorder the terms: 189.028 = 16.956r + 6.28r2 Solving 189.028 = 16.956r + 6.28r2 Solving for variable 'r'. Reorder the terms: 189.028 + -16.956r + -6.28r2 = 16.956r + -16.956r + 6.28r2 + -6.28r2 Combine like terms: 16.956r + -16.956r = 0.000 189.028 + -16.956r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 189.028 + -16.956r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 189.028 + -16.956r + -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'. -30.1 + 2.7r + r2 = 0 Move the constant term to the right: Add '30.1' to each side of the equation. -30.1 + 2.7r + 30.1 + r2 = 0 + 30.1 Reorder the terms: -30.1 + 30.1 + 2.7r + r2 = 0 + 30.1 Combine like terms: -30.1 + 30.1 = 0.0 0.0 + 2.7r + r2 = 0 + 30.1 2.7r + r2 = 0 + 30.1 Combine like terms: 0 + 30.1 = 30.1 2.7r + r2 = 30.1 The r term is 2.7r. Take half its coefficient (1.35). Square it (1.8225) and add it to both sides. Add '1.8225' to each side of the equation. 2.7r + 1.8225 + r2 = 30.1 + 1.8225 Reorder the terms: 1.8225 + 2.7r + r2 = 30.1 + 1.8225 Combine like terms: 30.1 + 1.8225 = 31.9225 1.8225 + 2.7r + r2 = 31.9225 Factor a perfect square on the left side: (r + 1.35)(r + 1.35) = 31.9225 Calculate the square root of the right side: 5.65 Break this problem into two subproblems by setting (r + 1.35) equal to 5.65 and -5.65.Subproblem 1
r + 1.35 = 5.65 Simplifying r + 1.35 = 5.65 Reorder the terms: 1.35 + r = 5.65 Solving 1.35 + r = 5.65 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + r = 5.65 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + r = 5.65 + -1.35 r = 5.65 + -1.35 Combine like terms: 5.65 + -1.35 = 4.3 r = 4.3 Simplifying r = 4.3Subproblem 2
r + 1.35 = -5.65 Simplifying r + 1.35 = -5.65 Reorder the terms: 1.35 + r = -5.65 Solving 1.35 + r = -5.65 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.35' to each side of the equation. 1.35 + -1.35 + r = -5.65 + -1.35 Combine like terms: 1.35 + -1.35 = 0.00 0.00 + r = -5.65 + -1.35 r = -5.65 + -1.35 Combine like terms: -5.65 + -1.35 = -7 r = -7 Simplifying r = -7Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.3, -7}
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