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Simplifying 8n2 + 3n + -13 = 0 Reorder the terms: -13 + 3n + 8n2 = 0 Solving -13 + 3n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.625 + 0.375n + n2 = 0 Move the constant term to the right: Add '1.625' to each side of the equation. -1.625 + 0.375n + 1.625 + n2 = 0 + 1.625 Reorder the terms: -1.625 + 1.625 + 0.375n + n2 = 0 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + 0.375n + n2 = 0 + 1.625 0.375n + n2 = 0 + 1.625 Combine like terms: 0 + 1.625 = 1.625 0.375n + n2 = 1.625 The n term is 0.375n. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375n + 0.03515625 + n2 = 1.625 + 0.03515625 Reorder the terms: 0.03515625 + 0.375n + n2 = 1.625 + 0.03515625 Combine like terms: 1.625 + 0.03515625 = 1.66015625 0.03515625 + 0.375n + n2 = 1.66015625 Factor a perfect square on the left side: (n + 0.1875)(n + 0.1875) = 1.66015625 Calculate the square root of the right side: 1.288470508 Break this problem into two subproblems by setting (n + 0.1875) equal to 1.288470508 and -1.288470508.Subproblem 1
n + 0.1875 = 1.288470508 Simplifying n + 0.1875 = 1.288470508 Reorder the terms: 0.1875 + n = 1.288470508 Solving 0.1875 + n = 1.288470508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + n = 1.288470508 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = 1.288470508 + -0.1875 n = 1.288470508 + -0.1875 Combine like terms: 1.288470508 + -0.1875 = 1.100970508 n = 1.100970508 Simplifying n = 1.100970508Subproblem 2
n + 0.1875 = -1.288470508 Simplifying n + 0.1875 = -1.288470508 Reorder the terms: 0.1875 + n = -1.288470508 Solving 0.1875 + n = -1.288470508 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + n = -1.288470508 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = -1.288470508 + -0.1875 n = -1.288470508 + -0.1875 Combine like terms: -1.288470508 + -0.1875 = -1.475970508 n = -1.475970508 Simplifying n = -1.475970508Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.100970508, -1.475970508}
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