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Simplifying n2 + n + -81 = 0 Reorder the terms: -81 + n + n2 = 0 Solving -81 + n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '81' to each side of the equation. -81 + n + 81 + n2 = 0 + 81 Reorder the terms: -81 + 81 + n + n2 = 0 + 81 Combine like terms: -81 + 81 = 0 0 + n + n2 = 0 + 81 n + n2 = 0 + 81 Combine like terms: 0 + 81 = 81 n + n2 = 81 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 81 + 0.25 Reorder the terms: 0.25 + n + n2 = 81 + 0.25 Combine like terms: 81 + 0.25 = 81.25 0.25 + n + n2 = 81.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 81.25 Calculate the square root of the right side: 9.013878189 Break this problem into two subproblems by setting (n + 0.5) equal to 9.013878189 and -9.013878189.Subproblem 1
n + 0.5 = 9.013878189 Simplifying n + 0.5 = 9.013878189 Reorder the terms: 0.5 + n = 9.013878189 Solving 0.5 + n = 9.013878189 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 9.013878189 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 9.013878189 + -0.5 n = 9.013878189 + -0.5 Combine like terms: 9.013878189 + -0.5 = 8.513878189 n = 8.513878189 Simplifying n = 8.513878189Subproblem 2
n + 0.5 = -9.013878189 Simplifying n + 0.5 = -9.013878189 Reorder the terms: 0.5 + n = -9.013878189 Solving 0.5 + n = -9.013878189 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -9.013878189 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -9.013878189 + -0.5 n = -9.013878189 + -0.5 Combine like terms: -9.013878189 + -0.5 = -9.513878189 n = -9.513878189 Simplifying n = -9.513878189Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.513878189, -9.513878189}
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