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Simplifying 8n(6n + 7) = 3 Reorder the terms: 8n(7 + 6n) = 3 (7 * 8n + 6n * 8n) = 3 (56n + 48n2) = 3 Solving 56n + 48n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + 56n + 48n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 56n + 48n2 = 0 Begin completing the square. Divide all terms by 48 the coefficient of the squared term: Divide each side by '48'. -0.0625 + 1.166666667n + n2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + 1.166666667n + 0.0625 + n2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + 1.166666667n + n2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + 1.166666667n + n2 = 0 + 0.0625 1.166666667n + n2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 1.166666667n + n2 = 0.0625 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 = 0.0625 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667n + n2 = 0.0625 + 0.3402777780 Combine like terms: 0.0625 + 0.3402777780 = 0.402777778 0.3402777780 + 1.166666667n + n2 = 0.402777778 Factor a perfect square on the left side: (n + 0.5833333335)(n + 0.5833333335) = 0.402777778 Calculate the square root of the right side: 0.634647759 Break this problem into two subproblems by setting (n + 0.5833333335) equal to 0.634647759 and -0.634647759.Subproblem 1
n + 0.5833333335 = 0.634647759 Simplifying n + 0.5833333335 = 0.634647759 Reorder the terms: 0.5833333335 + n = 0.634647759 Solving 0.5833333335 + n = 0.634647759 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 = 0.634647759 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = 0.634647759 + -0.5833333335 n = 0.634647759 + -0.5833333335 Combine like terms: 0.634647759 + -0.5833333335 = 0.0513144255 n = 0.0513144255 Simplifying n = 0.0513144255Subproblem 2
n + 0.5833333335 = -0.634647759 Simplifying n + 0.5833333335 = -0.634647759 Reorder the terms: 0.5833333335 + n = -0.634647759 Solving 0.5833333335 + n = -0.634647759 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 = -0.634647759 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + n = -0.634647759 + -0.5833333335 n = -0.634647759 + -0.5833333335 Combine like terms: -0.634647759 + -0.5833333335 = -1.2179810925 n = -1.2179810925 Simplifying n = -1.2179810925Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.0513144255, -1.2179810925}
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