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Simplifying (n + 11) * n = 84 Reorder the terms: (11 + n) * n = 84 Reorder the terms for easier multiplication: n(11 + n) = 84 (11 * n + n * n) = 84 (11n + n2) = 84 Solving 11n + n2 = 84 Solving for variable 'n'. Reorder the terms: -84 + 11n + n2 = 84 + -84 Combine like terms: 84 + -84 = 0 -84 + 11n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '84' to each side of the equation. -84 + 11n + 84 + n2 = 0 + 84 Reorder the terms: -84 + 84 + 11n + n2 = 0 + 84 Combine like terms: -84 + 84 = 0 0 + 11n + n2 = 0 + 84 11n + n2 = 0 + 84 Combine like terms: 0 + 84 = 84 11n + n2 = 84 The n term is 11n. Take half its coefficient (5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. 11n + 30.25 + n2 = 84 + 30.25 Reorder the terms: 30.25 + 11n + n2 = 84 + 30.25 Combine like terms: 84 + 30.25 = 114.25 30.25 + 11n + n2 = 114.25 Factor a perfect square on the left side: (n + 5.5)(n + 5.5) = 114.25 Calculate the square root of the right side: 10.688779163 Break this problem into two subproblems by setting (n + 5.5) equal to 10.688779163 and -10.688779163.Subproblem 1
n + 5.5 = 10.688779163 Simplifying n + 5.5 = 10.688779163 Reorder the terms: 5.5 + n = 10.688779163 Solving 5.5 + n = 10.688779163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = 10.688779163 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = 10.688779163 + -5.5 n = 10.688779163 + -5.5 Combine like terms: 10.688779163 + -5.5 = 5.188779163 n = 5.188779163 Simplifying n = 5.188779163Subproblem 2
n + 5.5 = -10.688779163 Simplifying n + 5.5 = -10.688779163 Reorder the terms: 5.5 + n = -10.688779163 Solving 5.5 + n = -10.688779163 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.5' to each side of the equation. 5.5 + -5.5 + n = -10.688779163 + -5.5 Combine like terms: 5.5 + -5.5 = 0.0 0.0 + n = -10.688779163 + -5.5 n = -10.688779163 + -5.5 Combine like terms: -10.688779163 + -5.5 = -16.188779163 n = -16.188779163 Simplifying n = -16.188779163Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.188779163, -16.188779163}
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