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Simplifying 2n2 + 8n + -46 = 0 Reorder the terms: -46 + 8n + 2n2 = 0 Solving -46 + 8n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-23 + 4n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-23 + 4n + n2)' equal to zero and attempt to solve: Simplifying -23 + 4n + n2 = 0 Solving -23 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 4n + 23 + n2 = 0 + 23 Reorder the terms: -23 + 23 + 4n + n2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 4n + n2 = 0 + 23 4n + n2 = 0 + 23 Combine like terms: 0 + 23 = 23 4n + n2 = 23 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 23 + 4 Reorder the terms: 4 + 4n + n2 = 23 + 4 Combine like terms: 23 + 4 = 27 4 + 4n + n2 = 27 Factor a perfect square on the left side: (n + 2)(n + 2) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (n + 2) equal to 5.196152423 and -5.196152423.Subproblem 1
n + 2 = 5.196152423 Simplifying n + 2 = 5.196152423 Reorder the terms: 2 + n = 5.196152423 Solving 2 + n = 5.196152423 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + n = 5.196152423 + -2 n = 5.196152423 + -2 Combine like terms: 5.196152423 + -2 = 3.196152423 n = 3.196152423 Simplifying n = 3.196152423Subproblem 2
n + 2 = -5.196152423 Simplifying n + 2 = -5.196152423 Reorder the terms: 2 + n = -5.196152423 Solving 2 + n = -5.196152423 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -5.196152423 + -2 Combine like terms: 2 + -2 = 0 0 + n = -5.196152423 + -2 n = -5.196152423 + -2 Combine like terms: -5.196152423 + -2 = -7.196152423 n = -7.196152423 Simplifying n = -7.196152423Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.196152423, -7.196152423}Solution
n = {3.196152423, -7.196152423}
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