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Simplifying 196r2 + -140r + 15 = 0 Reorder the terms: 15 + -140r + 196r2 = 0 Solving 15 + -140r + 196r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 196 the coefficient of the squared term: Divide each side by '196'. 0.07653061224 + -0.7142857143r + r2 = 0 Move the constant term to the right: Add '-0.07653061224' to each side of the equation. 0.07653061224 + -0.7142857143r + -0.07653061224 + r2 = 0 + -0.07653061224 Reorder the terms: 0.07653061224 + -0.07653061224 + -0.7142857143r + r2 = 0 + -0.07653061224 Combine like terms: 0.07653061224 + -0.07653061224 = 0.00000000000 0.00000000000 + -0.7142857143r + r2 = 0 + -0.07653061224 -0.7142857143r + r2 = 0 + -0.07653061224 Combine like terms: 0 + -0.07653061224 = -0.07653061224 -0.7142857143r + r2 = -0.07653061224 The r term is -0.7142857143r. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143r + 0.1275510204 + r2 = -0.07653061224 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143r + r2 = -0.07653061224 + 0.1275510204 Combine like terms: -0.07653061224 + 0.1275510204 = 0.05102040816 0.1275510204 + -0.7142857143r + r2 = 0.05102040816 Factor a perfect square on the left side: (r + -0.3571428572)(r + -0.3571428572) = 0.05102040816 Calculate the square root of the right side: 0.225876976 Break this problem into two subproblems by setting (r + -0.3571428572) equal to 0.225876976 and -0.225876976.Subproblem 1
r + -0.3571428572 = 0.225876976 Simplifying r + -0.3571428572 = 0.225876976 Reorder the terms: -0.3571428572 + r = 0.225876976 Solving -0.3571428572 + r = 0.225876976 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + r = 0.225876976 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + r = 0.225876976 + 0.3571428572 r = 0.225876976 + 0.3571428572 Combine like terms: 0.225876976 + 0.3571428572 = 0.5830198332 r = 0.5830198332 Simplifying r = 0.5830198332Subproblem 2
r + -0.3571428572 = -0.225876976 Simplifying r + -0.3571428572 = -0.225876976 Reorder the terms: -0.3571428572 + r = -0.225876976 Solving -0.3571428572 + r = -0.225876976 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + r = -0.225876976 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + r = -0.225876976 + 0.3571428572 r = -0.225876976 + 0.3571428572 Combine like terms: -0.225876976 + 0.3571428572 = 0.1312658812 r = 0.1312658812 Simplifying r = 0.1312658812Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.5830198332, 0.1312658812}
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