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Simplifying 8n2 + -16n + -51 = 0 Reorder the terms: -51 + -16n + 8n2 = 0 Solving -51 + -16n + 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'. -6.375 + -2n + n2 = 0 Move the constant term to the right: Add '6.375' to each side of the equation. -6.375 + -2n + 6.375 + n2 = 0 + 6.375 Reorder the terms: -6.375 + 6.375 + -2n + n2 = 0 + 6.375 Combine like terms: -6.375 + 6.375 = 0.000 0.000 + -2n + n2 = 0 + 6.375 -2n + n2 = 0 + 6.375 Combine like terms: 0 + 6.375 = 6.375 -2n + n2 = 6.375 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 6.375 + 1 Reorder the terms: 1 + -2n + n2 = 6.375 + 1 Combine like terms: 6.375 + 1 = 7.375 1 + -2n + n2 = 7.375 Factor a perfect square on the left side: (n + -1)(n + -1) = 7.375 Calculate the square root of the right side: 2.715695123 Break this problem into two subproblems by setting (n + -1) equal to 2.715695123 and -2.715695123.Subproblem 1
n + -1 = 2.715695123 Simplifying n + -1 = 2.715695123 Reorder the terms: -1 + n = 2.715695123 Solving -1 + n = 2.715695123 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 2.715695123 + 1 Combine like terms: -1 + 1 = 0 0 + n = 2.715695123 + 1 n = 2.715695123 + 1 Combine like terms: 2.715695123 + 1 = 3.715695123 n = 3.715695123 Simplifying n = 3.715695123Subproblem 2
n + -1 = -2.715695123 Simplifying n + -1 = -2.715695123 Reorder the terms: -1 + n = -2.715695123 Solving -1 + n = -2.715695123 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -2.715695123 + 1 Combine like terms: -1 + 1 = 0 0 + n = -2.715695123 + 1 n = -2.715695123 + 1 Combine like terms: -2.715695123 + 1 = -1.715695123 n = -1.715695123 Simplifying n = -1.715695123Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.715695123, -1.715695123}
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