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Simplifying 2708.815 = 6.28r2 + 240.524r Reorder the terms: 2708.815 = 240.524r + 6.28r2 Solving 2708.815 = 240.524r + 6.28r2 Solving for variable 'r'. Reorder the terms: 2708.815 + -240.524r + -6.28r2 = 240.524r + -240.524r + 6.28r2 + -6.28r2 Combine like terms: 240.524r + -240.524r = 0.000 2708.815 + -240.524r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2 2708.815 + -240.524r + -6.28r2 = 6.28r2 + -6.28r2 Combine like terms: 6.28r2 + -6.28r2 = 0.00 2708.815 + -240.524r + -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'. -431.3399682 + 38.3r + r2 = 0 Move the constant term to the right: Add '431.3399682' to each side of the equation. -431.3399682 + 38.3r + 431.3399682 + r2 = 0 + 431.3399682 Reorder the terms: -431.3399682 + 431.3399682 + 38.3r + r2 = 0 + 431.3399682 Combine like terms: -431.3399682 + 431.3399682 = 0.0000000 0.0000000 + 38.3r + r2 = 0 + 431.3399682 38.3r + r2 = 0 + 431.3399682 Combine like terms: 0 + 431.3399682 = 431.3399682 38.3r + r2 = 431.3399682 The r term is 38.3r. Take half its coefficient (19.15). Square it (366.7225) and add it to both sides. Add '366.7225' to each side of the equation. 38.3r + 366.7225 + r2 = 431.3399682 + 366.7225 Reorder the terms: 366.7225 + 38.3r + r2 = 431.3399682 + 366.7225 Combine like terms: 431.3399682 + 366.7225 = 798.0624682 366.7225 + 38.3r + r2 = 798.0624682 Factor a perfect square on the left side: (r + 19.15)(r + 19.15) = 798.0624682 Calculate the square root of the right side: 28.249999437 Break this problem into two subproblems by setting (r + 19.15) equal to 28.249999437 and -28.249999437.Subproblem 1
r + 19.15 = 28.249999437 Simplifying r + 19.15 = 28.249999437 Reorder the terms: 19.15 + r = 28.249999437 Solving 19.15 + r = 28.249999437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-19.15' to each side of the equation. 19.15 + -19.15 + r = 28.249999437 + -19.15 Combine like terms: 19.15 + -19.15 = 0.00 0.00 + r = 28.249999437 + -19.15 r = 28.249999437 + -19.15 Combine like terms: 28.249999437 + -19.15 = 9.099999437 r = 9.099999437 Simplifying r = 9.099999437Subproblem 2
r + 19.15 = -28.249999437 Simplifying r + 19.15 = -28.249999437 Reorder the terms: 19.15 + r = -28.249999437 Solving 19.15 + r = -28.249999437 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-19.15' to each side of the equation. 19.15 + -19.15 + r = -28.249999437 + -19.15 Combine like terms: 19.15 + -19.15 = 0.00 0.00 + r = -28.249999437 + -19.15 r = -28.249999437 + -19.15 Combine like terms: -28.249999437 + -19.15 = -47.399999437 r = -47.399999437 Simplifying r = -47.399999437Solution
The solution to the problem is based on the solutions from the subproblems. r = {9.099999437, -47.399999437}
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