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Simplifying 8n2 + 3n + -1 = 0 Reorder the terms: -1 + 3n + 8n2 = 0 Solving -1 + 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'. -0.125 + 0.375n + n2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + 0.375n + 0.125 + n2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + 0.375n + n2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + 0.375n + n2 = 0 + 0.125 0.375n + n2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 0.375n + n2 = 0.125 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 = 0.125 + 0.03515625 Reorder the terms: 0.03515625 + 0.375n + n2 = 0.125 + 0.03515625 Combine like terms: 0.125 + 0.03515625 = 0.16015625 0.03515625 + 0.375n + n2 = 0.16015625 Factor a perfect square on the left side: (n + 0.1875)(n + 0.1875) = 0.16015625 Calculate the square root of the right side: 0.400195265 Break this problem into two subproblems by setting (n + 0.1875) equal to 0.400195265 and -0.400195265.Subproblem 1
n + 0.1875 = 0.400195265 Simplifying n + 0.1875 = 0.400195265 Reorder the terms: 0.1875 + n = 0.400195265 Solving 0.1875 + n = 0.400195265 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 = 0.400195265 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = 0.400195265 + -0.1875 n = 0.400195265 + -0.1875 Combine like terms: 0.400195265 + -0.1875 = 0.212695265 n = 0.212695265 Simplifying n = 0.212695265Subproblem 2
n + 0.1875 = -0.400195265 Simplifying n + 0.1875 = -0.400195265 Reorder the terms: 0.1875 + n = -0.400195265 Solving 0.1875 + n = -0.400195265 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 = -0.400195265 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + n = -0.400195265 + -0.1875 n = -0.400195265 + -0.1875 Combine like terms: -0.400195265 + -0.1875 = -0.587695265 n = -0.587695265 Simplifying n = -0.587695265Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.212695265, -0.587695265}
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