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Simplifying 8n2 = -6n + 11 Reorder the terms: 8n2 = 11 + -6n Solving 8n2 = 11 + -6n Solving for variable 'n'. Reorder the terms: -11 + 6n + 8n2 = 11 + -6n + -11 + 6n Reorder the terms: -11 + 6n + 8n2 = 11 + -11 + -6n + 6n Combine like terms: 11 + -11 = 0 -11 + 6n + 8n2 = 0 + -6n + 6n -11 + 6n + 8n2 = -6n + 6n Combine like terms: -6n + 6n = 0 -11 + 6n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.375 + 0.75n + n2 = 0 Move the constant term to the right: Add '1.375' to each side of the equation. -1.375 + 0.75n + 1.375 + n2 = 0 + 1.375 Reorder the terms: -1.375 + 1.375 + 0.75n + n2 = 0 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + 0.75n + n2 = 0 + 1.375 0.75n + n2 = 0 + 1.375 Combine like terms: 0 + 1.375 = 1.375 0.75n + n2 = 1.375 The n term is 0.75n. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75n + 0.140625 + n2 = 1.375 + 0.140625 Reorder the terms: 0.140625 + 0.75n + n2 = 1.375 + 0.140625 Combine like terms: 1.375 + 0.140625 = 1.515625 0.140625 + 0.75n + n2 = 1.515625 Factor a perfect square on the left side: (n + 0.375)(n + 0.375) = 1.515625 Calculate the square root of the right side: 1.231107225 Break this problem into two subproblems by setting (n + 0.375) equal to 1.231107225 and -1.231107225.Subproblem 1
n + 0.375 = 1.231107225 Simplifying n + 0.375 = 1.231107225 Reorder the terms: 0.375 + n = 1.231107225 Solving 0.375 + n = 1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + n = 1.231107225 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = 1.231107225 + -0.375 n = 1.231107225 + -0.375 Combine like terms: 1.231107225 + -0.375 = 0.856107225 n = 0.856107225 Simplifying n = 0.856107225Subproblem 2
n + 0.375 = -1.231107225 Simplifying n + 0.375 = -1.231107225 Reorder the terms: 0.375 + n = -1.231107225 Solving 0.375 + n = -1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + n = -1.231107225 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = -1.231107225 + -0.375 n = -1.231107225 + -0.375 Combine like terms: -1.231107225 + -0.375 = -1.606107225 n = -1.606107225 Simplifying n = -1.606107225Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.856107225, -1.606107225}
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