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Simplifying n2 + -23 = -8n Reorder the terms: -23 + n2 = -8n Solving -23 + n2 = -8n Solving for variable 'n'. Reorder the terms: -23 + 8n + n2 = -8n + 8n Combine like terms: -8n + 8n = 0 -23 + 8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 8n + 23 + n2 = 0 + 23 Reorder the terms: -23 + 23 + 8n + n2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 8n + n2 = 0 + 23 8n + n2 = 0 + 23 Combine like terms: 0 + 23 = 23 8n + n2 = 23 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 = 23 + 16 Reorder the terms: 16 + 8n + n2 = 23 + 16 Combine like terms: 23 + 16 = 39 16 + 8n + n2 = 39 Factor a perfect square on the left side: (n + 4)(n + 4) = 39 Calculate the square root of the right side: 6.244997998 Break this problem into two subproblems by setting (n + 4) equal to 6.244997998 and -6.244997998.Subproblem 1
n + 4 = 6.244997998 Simplifying n + 4 = 6.244997998 Reorder the terms: 4 + n = 6.244997998 Solving 4 + n = 6.244997998 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 = 6.244997998 + -4 Combine like terms: 4 + -4 = 0 0 + n = 6.244997998 + -4 n = 6.244997998 + -4 Combine like terms: 6.244997998 + -4 = 2.244997998 n = 2.244997998 Simplifying n = 2.244997998Subproblem 2
n + 4 = -6.244997998 Simplifying n + 4 = -6.244997998 Reorder the terms: 4 + n = -6.244997998 Solving 4 + n = -6.244997998 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 = -6.244997998 + -4 Combine like terms: 4 + -4 = 0 0 + n = -6.244997998 + -4 n = -6.244997998 + -4 Combine like terms: -6.244997998 + -4 = -10.244997998 n = -10.244997998 Simplifying n = -10.244997998Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.244997998, -10.244997998}
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