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Simplifying 6n2 + 7n + -7 = 0 Reorder the terms: -7 + 7n + 6n2 = 0 Solving -7 + 7n + 6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + 1.166666667n + n2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667n + 1.166666667 + n2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 1.166666667n + n2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 1.166666667n + n2 = 0 + 1.166666667 1.166666667n + n2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 1.166666667n + n2 = 1.166666667 The n term is 1.166666667n. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667n + 0.3402777780 + n2 = 1.166666667 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667n + n2 = 1.166666667 + 0.3402777780 Combine like terms: 1.166666667 + 0.3402777780 = 1.506944445 0.3402777780 + 1.166666667n + n2 = 1.506944445 Factor a perfect square on the left side: (n + 0.5833333335)(n + 0.5833333335) = 1.506944445 Calculate the square root of the right side: 1.227576655 Break this problem into two subproblems by setting (n + 0.5833333335) equal to 1.227576655 and -1.227576655.Subproblem 1
n + 0.5833333335 = 1.227576655 Simplifying n + 0.5833333335 = 1.227576655 Reorder the terms: 0.5833333335 + n = 1.227576655 Solving 0.5833333335 + n = 1.227576655 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = 1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = 1.227576655 + -0.5833333335 n = 1.227576655 + -0.5833333335 Combine like terms: 1.227576655 + -0.5833333335 = 0.6442433215 n = 0.6442433215 Simplifying n = 0.6442433215Subproblem 2
n + 0.5833333335 = -1.227576655 Simplifying n + 0.5833333335 = -1.227576655 Reorder the terms: 0.5833333335 + n = -1.227576655 Solving 0.5833333335 + n = -1.227576655 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + n = -1.227576655 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = -1.227576655 + -0.5833333335 n = -1.227576655 + -0.5833333335 Combine like terms: -1.227576655 + -0.5833333335 = -1.8109099885 n = -1.8109099885 Simplifying n = -1.8109099885Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.6442433215, -1.8109099885}
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