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Simplifying 2(1 + 4n) * 8n = 82 Reorder the terms for easier multiplication: 2 * 8n(1 + 4n) = 82 Multiply 2 * 8 16n(1 + 4n) = 82 (1 * 16n + 4n * 16n) = 82 (16n + 64n2) = 82 Solving 16n + 64n2 = 82 Solving for variable 'n'. Reorder the terms: -82 + 16n + 64n2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + 16n + 64n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-41 + 8n + 32n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-41 + 8n + 32n2)' equal to zero and attempt to solve: Simplifying -41 + 8n + 32n2 = 0 Solving -41 + 8n + 32n2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -1.28125 + 0.25n + n2 = 0 Move the constant term to the right: Add '1.28125' to each side of the equation. -1.28125 + 0.25n + 1.28125 + n2 = 0 + 1.28125 Reorder the terms: -1.28125 + 1.28125 + 0.25n + n2 = 0 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + 0.25n + n2 = 0 + 1.28125 0.25n + n2 = 0 + 1.28125 Combine like terms: 0 + 1.28125 = 1.28125 0.25n + n2 = 1.28125 The n term is 0.25n. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25n + 0.015625 + n2 = 1.28125 + 0.015625 Reorder the terms: 0.015625 + 0.25n + n2 = 1.28125 + 0.015625 Combine like terms: 1.28125 + 0.015625 = 1.296875 0.015625 + 0.25n + n2 = 1.296875 Factor a perfect square on the left side: (n + 0.125)(n + 0.125) = 1.296875 Calculate the square root of the right side: 1.138804197 Break this problem into two subproblems by setting (n + 0.125) equal to 1.138804197 and -1.138804197.Subproblem 1
n + 0.125 = 1.138804197 Simplifying n + 0.125 = 1.138804197 Reorder the terms: 0.125 + n = 1.138804197 Solving 0.125 + n = 1.138804197 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + n = 1.138804197 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + n = 1.138804197 + -0.125 n = 1.138804197 + -0.125 Combine like terms: 1.138804197 + -0.125 = 1.013804197 n = 1.013804197 Simplifying n = 1.013804197Subproblem 2
n + 0.125 = -1.138804197 Simplifying n + 0.125 = -1.138804197 Reorder the terms: 0.125 + n = -1.138804197 Solving 0.125 + n = -1.138804197 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + n = -1.138804197 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + n = -1.138804197 + -0.125 n = -1.138804197 + -0.125 Combine like terms: -1.138804197 + -0.125 = -1.263804197 n = -1.263804197 Simplifying n = -1.263804197Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.013804197, -1.263804197}Solution
n = {1.013804197, -1.263804197}
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