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Simplifying 2n2 + 16n + 34 = 6 Reorder the terms: 34 + 16n + 2n2 = 6 Solving 34 + 16n + 2n2 = 6 Solving for variable 'n'. Reorder the terms: 34 + -6 + 16n + 2n2 = 6 + -6 Combine like terms: 34 + -6 = 28 28 + 16n + 2n2 = 6 + -6 Combine like terms: 6 + -6 = 0 28 + 16n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(14 + 8n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(14 + 8n + n2)' equal to zero and attempt to solve: Simplifying 14 + 8n + n2 = 0 Solving 14 + 8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-14' to each side of the equation. 14 + 8n + -14 + n2 = 0 + -14 Reorder the terms: 14 + -14 + 8n + n2 = 0 + -14 Combine like terms: 14 + -14 = 0 0 + 8n + n2 = 0 + -14 8n + n2 = 0 + -14 Combine like terms: 0 + -14 = -14 8n + n2 = -14 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 = -14 + 16 Reorder the terms: 16 + 8n + n2 = -14 + 16 Combine like terms: -14 + 16 = 2 16 + 8n + n2 = 2 Factor a perfect square on the left side: (n + 4)(n + 4) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (n + 4) equal to 1.414213562 and -1.414213562.Subproblem 1
n + 4 = 1.414213562 Simplifying n + 4 = 1.414213562 Reorder the terms: 4 + n = 1.414213562 Solving 4 + n = 1.414213562 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 = 1.414213562 + -4 Combine like terms: 4 + -4 = 0 0 + n = 1.414213562 + -4 n = 1.414213562 + -4 Combine like terms: 1.414213562 + -4 = -2.585786438 n = -2.585786438 Simplifying n = -2.585786438Subproblem 2
n + 4 = -1.414213562 Simplifying n + 4 = -1.414213562 Reorder the terms: 4 + n = -1.414213562 Solving 4 + n = -1.414213562 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 = -1.414213562 + -4 Combine like terms: 4 + -4 = 0 0 + n = -1.414213562 + -4 n = -1.414213562 + -4 Combine like terms: -1.414213562 + -4 = -5.414213562 n = -5.414213562 Simplifying n = -5.414213562Solution
The solution to the problem is based on the solutions from the subproblems. n = {-2.585786438, -5.414213562}Solution
n = {-2.585786438, -5.414213562}
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