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Simplifying 8n2 + -4n + 2 = 5n + 11 Reorder the terms: 2 + -4n + 8n2 = 5n + 11 Reorder the terms: 2 + -4n + 8n2 = 11 + 5n Solving 2 + -4n + 8n2 = 11 + 5n Solving for variable 'n'. Reorder the terms: 2 + -11 + -4n + -5n + 8n2 = 11 + 5n + -11 + -5n Combine like terms: 2 + -11 = -9 -9 + -4n + -5n + 8n2 = 11 + 5n + -11 + -5n Combine like terms: -4n + -5n = -9n -9 + -9n + 8n2 = 11 + 5n + -11 + -5n Reorder the terms: -9 + -9n + 8n2 = 11 + -11 + 5n + -5n Combine like terms: 11 + -11 = 0 -9 + -9n + 8n2 = 0 + 5n + -5n -9 + -9n + 8n2 = 5n + -5n Combine like terms: 5n + -5n = 0 -9 + -9n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.125 + -1.125n + n2 = 0 Move the constant term to the right: Add '1.125' to each side of the equation. -1.125 + -1.125n + 1.125 + n2 = 0 + 1.125 Reorder the terms: -1.125 + 1.125 + -1.125n + n2 = 0 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + -1.125n + n2 = 0 + 1.125 -1.125n + n2 = 0 + 1.125 Combine like terms: 0 + 1.125 = 1.125 -1.125n + n2 = 1.125 The n term is -1.125n. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125n + 0.31640625 + n2 = 1.125 + 0.31640625 Reorder the terms: 0.31640625 + -1.125n + n2 = 1.125 + 0.31640625 Combine like terms: 1.125 + 0.31640625 = 1.44140625 0.31640625 + -1.125n + n2 = 1.44140625 Factor a perfect square on the left side: (n + -0.5625)(n + -0.5625) = 1.44140625 Calculate the square root of the right side: 1.200585795 Break this problem into two subproblems by setting (n + -0.5625) equal to 1.200585795 and -1.200585795.Subproblem 1
n + -0.5625 = 1.200585795 Simplifying n + -0.5625 = 1.200585795 Reorder the terms: -0.5625 + n = 1.200585795 Solving -0.5625 + n = 1.200585795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = 1.200585795 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = 1.200585795 + 0.5625 n = 1.200585795 + 0.5625 Combine like terms: 1.200585795 + 0.5625 = 1.763085795 n = 1.763085795 Simplifying n = 1.763085795Subproblem 2
n + -0.5625 = -1.200585795 Simplifying n + -0.5625 = -1.200585795 Reorder the terms: -0.5625 + n = -1.200585795 Solving -0.5625 + n = -1.200585795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + n = -1.200585795 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + n = -1.200585795 + 0.5625 n = -1.200585795 + 0.5625 Combine like terms: -1.200585795 + 0.5625 = -0.638085795 n = -0.638085795 Simplifying n = -0.638085795Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.763085795, -0.638085795}
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