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Simplifying 3 + 8n = 3n(2n + -1) Reorder the terms: 3 + 8n = 3n(-1 + 2n) 3 + 8n = (-1 * 3n + 2n * 3n) 3 + 8n = (-3n + 6n2) Solving 3 + 8n = -3n + 6n2 Solving for variable 'n'. Combine like terms: 8n + 3n = 11n 3 + 11n + -6n2 = -3n + 6n2 + 3n + -6n2 Reorder the terms: 3 + 11n + -6n2 = -3n + 3n + 6n2 + -6n2 Combine like terms: -3n + 3n = 0 3 + 11n + -6n2 = 0 + 6n2 + -6n2 3 + 11n + -6n2 = 6n2 + -6n2 Combine like terms: 6n2 + -6n2 = 0 3 + 11n + -6n2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.5 + -1.833333333n + n2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.833333333n + 0.5 + n2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.833333333n + n2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.833333333n + n2 = 0 + 0.5 -1.833333333n + n2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.833333333n + n2 = 0.5 The n term is -1.833333333n. Take half its coefficient (-0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. -1.833333333n + 0.8402777775 + n2 = 0.5 + 0.8402777775 Reorder the terms: 0.8402777775 + -1.833333333n + n2 = 0.5 + 0.8402777775 Combine like terms: 0.5 + 0.8402777775 = 1.3402777775 0.8402777775 + -1.833333333n + n2 = 1.3402777775 Factor a perfect square on the left side: (n + -0.9166666665)(n + -0.9166666665) = 1.3402777775 Calculate the square root of the right side: 1.157703666 Break this problem into two subproblems by setting (n + -0.9166666665) equal to 1.157703666 and -1.157703666.Subproblem 1
n + -0.9166666665 = 1.157703666 Simplifying n + -0.9166666665 = 1.157703666 Reorder the terms: -0.9166666665 + n = 1.157703666 Solving -0.9166666665 + n = 1.157703666 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + n = 1.157703666 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + n = 1.157703666 + 0.9166666665 n = 1.157703666 + 0.9166666665 Combine like terms: 1.157703666 + 0.9166666665 = 2.0743703325 n = 2.0743703325 Simplifying n = 2.0743703325Subproblem 2
n + -0.9166666665 = -1.157703666 Simplifying n + -0.9166666665 = -1.157703666 Reorder the terms: -0.9166666665 + n = -1.157703666 Solving -0.9166666665 + n = -1.157703666 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9166666665' to each side of the equation. -0.9166666665 + 0.9166666665 + n = -1.157703666 + 0.9166666665 Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000 0.0000000000 + n = -1.157703666 + 0.9166666665 n = -1.157703666 + 0.9166666665 Combine like terms: -1.157703666 + 0.9166666665 = -0.2410369995 n = -0.2410369995 Simplifying n = -0.2410369995Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.0743703325, -0.2410369995}
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