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Simplifying 6(n + 8) * n = 5.6 Reorder the terms: 6(8 + n) * n = 5.6 Reorder the terms for easier multiplication: 6n(8 + n) = 5.6 (8 * 6n + n * 6n) = 5.6 (48n + 6n2) = 5.6 Solving 48n + 6n2 = 5.6 Solving for variable 'n'. Reorder the terms: -5.6 + 48n + 6n2 = 5.6 + -5.6 Combine like terms: 5.6 + -5.6 = 0.0 -5.6 + 48n + 6n2 = 0.0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.9333333333 + 8n + n2 = 0.0 Move the constant term to the right: Add '0.9333333333' to each side of the equation. -0.9333333333 + 8n + 0.9333333333 + n2 = 0.0 + 0.9333333333 Reorder the terms: -0.9333333333 + 0.9333333333 + 8n + n2 = 0.0 + 0.9333333333 Combine like terms: -0.9333333333 + 0.9333333333 = 0.0000000000 0.0000000000 + 8n + n2 = 0.0 + 0.9333333333 8n + n2 = 0.0 + 0.9333333333 Combine like terms: 0.0 + 0.9333333333 = 0.9333333333 8n + n2 = 0.9333333333 The n term is 8n. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8n + 16 + n2 = 0.9333333333 + 16 Reorder the terms: 16 + 8n + n2 = 0.9333333333 + 16 Combine like terms: 0.9333333333 + 16 = 16.9333333333 16 + 8n + n2 = 16.9333333333 Factor a perfect square on the left side: (n + 4)(n + 4) = 16.9333333333 Calculate the square root of the right side: 4.115013163 Break this problem into two subproblems by setting (n + 4) equal to 4.115013163 and -4.115013163.Subproblem 1
n + 4 = 4.115013163 Simplifying n + 4 = 4.115013163 Reorder the terms: 4 + n = 4.115013163 Solving 4 + n = 4.115013163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = 4.115013163 + -4 Combine like terms: 4 + -4 = 0 0 + n = 4.115013163 + -4 n = 4.115013163 + -4 Combine like terms: 4.115013163 + -4 = 0.115013163 n = 0.115013163 Simplifying n = 0.115013163Subproblem 2
n + 4 = -4.115013163 Simplifying n + 4 = -4.115013163 Reorder the terms: 4 + n = -4.115013163 Solving 4 + n = -4.115013163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = -4.115013163 + -4 Combine like terms: 4 + -4 = 0 0 + n = -4.115013163 + -4 n = -4.115013163 + -4 Combine like terms: -4.115013163 + -4 = -8.115013163 n = -8.115013163 Simplifying n = -8.115013163Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.115013163, -8.115013163}
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