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Simplifying 8n(6n + 5) = 3 Reorder the terms: 8n(5 + 6n) = 3 (5 * 8n + 6n * 8n) = 3 (40n + 48n2) = 3 Solving 40n + 48n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + 40n + 48n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 40n + 48n2 = 0 Begin completing the square. Divide all terms by 48 the coefficient of the squared term: Divide each side by '48'. -0.0625 + 0.8333333333n + n2 = 0 Move the constant term to the right: Add '0.0625' to each side of the equation. -0.0625 + 0.8333333333n + 0.0625 + n2 = 0 + 0.0625 Reorder the terms: -0.0625 + 0.0625 + 0.8333333333n + n2 = 0 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + 0.8333333333n + n2 = 0 + 0.0625 0.8333333333n + n2 = 0 + 0.0625 Combine like terms: 0 + 0.0625 = 0.0625 0.8333333333n + n2 = 0.0625 The n term is 0.8333333333n. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333n + 0.1736111111 + n2 = 0.0625 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333n + n2 = 0.0625 + 0.1736111111 Combine like terms: 0.0625 + 0.1736111111 = 0.2361111111 0.1736111111 + 0.8333333333n + n2 = 0.2361111111 Factor a perfect square on the left side: (n + 0.4166666667)(n + 0.4166666667) = 0.2361111111 Calculate the square root of the right side: 0.485912658 Break this problem into two subproblems by setting (n + 0.4166666667) equal to 0.485912658 and -0.485912658.Subproblem 1
n + 0.4166666667 = 0.485912658 Simplifying n + 0.4166666667 = 0.485912658 Reorder the terms: 0.4166666667 + n = 0.485912658 Solving 0.4166666667 + n = 0.485912658 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = 0.485912658 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = 0.485912658 + -0.4166666667 n = 0.485912658 + -0.4166666667 Combine like terms: 0.485912658 + -0.4166666667 = 0.0692459913 n = 0.0692459913 Simplifying n = 0.0692459913Subproblem 2
n + 0.4166666667 = -0.485912658 Simplifying n + 0.4166666667 = -0.485912658 Reorder the terms: 0.4166666667 + n = -0.485912658 Solving 0.4166666667 + n = -0.485912658 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + n = -0.485912658 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + n = -0.485912658 + -0.4166666667 n = -0.485912658 + -0.4166666667 Combine like terms: -0.485912658 + -0.4166666667 = -0.9025793247 n = -0.9025793247 Simplifying n = -0.9025793247Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.0692459913, -0.9025793247}
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