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Simplifying 7r2 + 5r + -11 = 0 Reorder the terms: -11 + 5r + 7r2 = 0 Solving -11 + 5r + 7r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.571428571 + 0.7142857143r + r2 = 0 Move the constant term to the right: Add '1.571428571' to each side of the equation. -1.571428571 + 0.7142857143r + 1.571428571 + r2 = 0 + 1.571428571 Reorder the terms: -1.571428571 + 1.571428571 + 0.7142857143r + r2 = 0 + 1.571428571 Combine like terms: -1.571428571 + 1.571428571 = 0.000000000 0.000000000 + 0.7142857143r + r2 = 0 + 1.571428571 0.7142857143r + r2 = 0 + 1.571428571 Combine like terms: 0 + 1.571428571 = 1.571428571 0.7142857143r + r2 = 1.571428571 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 = 1.571428571 + 0.1275510204 Reorder the terms: 0.1275510204 + 0.7142857143r + r2 = 1.571428571 + 0.1275510204 Combine like terms: 1.571428571 + 0.1275510204 = 1.6989795914 0.1275510204 + 0.7142857143r + r2 = 1.6989795914 Factor a perfect square on the left side: (r + 0.3571428572)(r + 0.3571428572) = 1.6989795914 Calculate the square root of the right side: 1.303449113 Break this problem into two subproblems by setting (r + 0.3571428572) equal to 1.303449113 and -1.303449113.Subproblem 1
r + 0.3571428572 = 1.303449113 Simplifying r + 0.3571428572 = 1.303449113 Reorder the terms: 0.3571428572 + r = 1.303449113 Solving 0.3571428572 + r = 1.303449113 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 = 1.303449113 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + r = 1.303449113 + -0.3571428572 r = 1.303449113 + -0.3571428572 Combine like terms: 1.303449113 + -0.3571428572 = 0.9463062558 r = 0.9463062558 Simplifying r = 0.9463062558Subproblem 2
r + 0.3571428572 = -1.303449113 Simplifying r + 0.3571428572 = -1.303449113 Reorder the terms: 0.3571428572 + r = -1.303449113 Solving 0.3571428572 + r = -1.303449113 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 = -1.303449113 + -0.3571428572 Combine like terms: 0.3571428572 + -0.3571428572 = 0.0000000000 0.0000000000 + r = -1.303449113 + -0.3571428572 r = -1.303449113 + -0.3571428572 Combine like terms: -1.303449113 + -0.3571428572 = -1.6605919702 r = -1.6605919702 Simplifying r = -1.6605919702Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.9463062558, -1.6605919702}
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