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Simplifying n2 + -18n + 31 = 0 Reorder the terms: 31 + -18n + n2 = 0 Solving 31 + -18n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '-31' to each side of the equation. 31 + -18n + -31 + n2 = 0 + -31 Reorder the terms: 31 + -31 + -18n + n2 = 0 + -31 Combine like terms: 31 + -31 = 0 0 + -18n + n2 = 0 + -31 -18n + n2 = 0 + -31 Combine like terms: 0 + -31 = -31 -18n + n2 = -31 The n term is -18n. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18n + 81 + n2 = -31 + 81 Reorder the terms: 81 + -18n + n2 = -31 + 81 Combine like terms: -31 + 81 = 50 81 + -18n + n2 = 50 Factor a perfect square on the left side: (n + -9)(n + -9) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (n + -9) equal to 7.071067812 and -7.071067812.Subproblem 1
n + -9 = 7.071067812 Simplifying n + -9 = 7.071067812 Reorder the terms: -9 + n = 7.071067812 Solving -9 + n = 7.071067812 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 7.071067812 + 9 Combine like terms: -9 + 9 = 0 0 + n = 7.071067812 + 9 n = 7.071067812 + 9 Combine like terms: 7.071067812 + 9 = 16.071067812 n = 16.071067812 Simplifying n = 16.071067812Subproblem 2
n + -9 = -7.071067812 Simplifying n + -9 = -7.071067812 Reorder the terms: -9 + n = -7.071067812 Solving -9 + n = -7.071067812 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -7.071067812 + 9 Combine like terms: -9 + 9 = 0 0 + n = -7.071067812 + 9 n = -7.071067812 + 9 Combine like terms: -7.071067812 + 9 = 1.928932188 n = 1.928932188 Simplifying n = 1.928932188Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.071067812, 1.928932188}
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