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Simplifying 2n2 = -8n + -5 Reorder the terms: 2n2 = -5 + -8n Solving 2n2 = -5 + -8n Solving for variable 'n'. Reorder the terms: 5 + 8n + 2n2 = -5 + -8n + 5 + 8n Reorder the terms: 5 + 8n + 2n2 = -5 + 5 + -8n + 8n Combine like terms: -5 + 5 = 0 5 + 8n + 2n2 = 0 + -8n + 8n 5 + 8n + 2n2 = -8n + 8n Combine like terms: -8n + 8n = 0 5 + 8n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + 4n + n2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + 4n + -2.5 + n2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + 4n + n2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + 4n + n2 = 0 + -2.5 4n + n2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 4n + n2 = -2.5 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = -2.5 + 4 Reorder the terms: 4 + 4n + n2 = -2.5 + 4 Combine like terms: -2.5 + 4 = 1.5 4 + 4n + n2 = 1.5 Factor a perfect square on the left side: (n + 2)(n + 2) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (n + 2) equal to 1.224744871 and -1.224744871.Subproblem 1
n + 2 = 1.224744871 Simplifying n + 2 = 1.224744871 Reorder the terms: 2 + n = 1.224744871 Solving 2 + n = 1.224744871 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 1.224744871 + -2 Combine like terms: 2 + -2 = 0 0 + n = 1.224744871 + -2 n = 1.224744871 + -2 Combine like terms: 1.224744871 + -2 = -0.775255129 n = -0.775255129 Simplifying n = -0.775255129Subproblem 2
n + 2 = -1.224744871 Simplifying n + 2 = -1.224744871 Reorder the terms: 2 + n = -1.224744871 Solving 2 + n = -1.224744871 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -1.224744871 + -2 Combine like terms: 2 + -2 = 0 0 + n = -1.224744871 + -2 n = -1.224744871 + -2 Combine like terms: -1.224744871 + -2 = -3.224744871 n = -3.224744871 Simplifying n = -3.224744871Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.775255129, -3.224744871}
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