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Simplifying 11n2 + 4n + -16 = -6 Reorder the terms: -16 + 4n + 11n2 = -6 Solving -16 + 4n + 11n2 = -6 Solving for variable 'n'. Reorder the terms: -16 + 6 + 4n + 11n2 = -6 + 6 Combine like terms: -16 + 6 = -10 -10 + 4n + 11n2 = -6 + 6 Combine like terms: -6 + 6 = 0 -10 + 4n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.9090909091 + 0.3636363636n + n2 = 0 Move the constant term to the right: Add '0.9090909091' to each side of the equation. -0.9090909091 + 0.3636363636n + 0.9090909091 + n2 = 0 + 0.9090909091 Reorder the terms: -0.9090909091 + 0.9090909091 + 0.3636363636n + n2 = 0 + 0.9090909091 Combine like terms: -0.9090909091 + 0.9090909091 = 0.0000000000 0.0000000000 + 0.3636363636n + n2 = 0 + 0.9090909091 0.3636363636n + n2 = 0 + 0.9090909091 Combine like terms: 0 + 0.9090909091 = 0.9090909091 0.3636363636n + n2 = 0.9090909091 The n term is 0.3636363636n. Take half its coefficient (0.1818181818). Square it (0.03305785123) and add it to both sides. Add '0.03305785123' to each side of the equation. 0.3636363636n + 0.03305785123 + n2 = 0.9090909091 + 0.03305785123 Reorder the terms: 0.03305785123 + 0.3636363636n + n2 = 0.9090909091 + 0.03305785123 Combine like terms: 0.9090909091 + 0.03305785123 = 0.94214876033 0.03305785123 + 0.3636363636n + n2 = 0.94214876033 Factor a perfect square on the left side: (n + 0.1818181818)(n + 0.1818181818) = 0.94214876033 Calculate the square root of the right side: 0.970643477 Break this problem into two subproblems by setting (n + 0.1818181818) equal to 0.970643477 and -0.970643477.Subproblem 1
n + 0.1818181818 = 0.970643477 Simplifying n + 0.1818181818 = 0.970643477 Reorder the terms: 0.1818181818 + n = 0.970643477 Solving 0.1818181818 + n = 0.970643477 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1818181818' to each side of the equation. 0.1818181818 + -0.1818181818 + n = 0.970643477 + -0.1818181818 Combine like terms: 0.1818181818 + -0.1818181818 = 0.0000000000 0.0000000000 + n = 0.970643477 + -0.1818181818 n = 0.970643477 + -0.1818181818 Combine like terms: 0.970643477 + -0.1818181818 = 0.7888252952 n = 0.7888252952 Simplifying n = 0.7888252952Subproblem 2
n + 0.1818181818 = -0.970643477 Simplifying n + 0.1818181818 = -0.970643477 Reorder the terms: 0.1818181818 + n = -0.970643477 Solving 0.1818181818 + n = -0.970643477 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1818181818' to each side of the equation. 0.1818181818 + -0.1818181818 + n = -0.970643477 + -0.1818181818 Combine like terms: 0.1818181818 + -0.1818181818 = 0.0000000000 0.0000000000 + n = -0.970643477 + -0.1818181818 n = -0.970643477 + -0.1818181818 Combine like terms: -0.970643477 + -0.1818181818 = -1.1524616588 n = -1.1524616588 Simplifying n = -1.1524616588Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.7888252952, -1.1524616588}
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