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Simplifying n2 + -9n + -31 = 0 Reorder the terms: -31 + -9n + n2 = 0 Solving -31 + -9n + 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 + -9n + 31 + n2 = 0 + 31 Reorder the terms: -31 + 31 + -9n + n2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + -9n + n2 = 0 + 31 -9n + n2 = 0 + 31 Combine like terms: 0 + 31 = 31 -9n + n2 = 31 The n term is -9n. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9n + 20.25 + n2 = 31 + 20.25 Reorder the terms: 20.25 + -9n + n2 = 31 + 20.25 Combine like terms: 31 + 20.25 = 51.25 20.25 + -9n + n2 = 51.25 Factor a perfect square on the left side: (n + -4.5)(n + -4.5) = 51.25 Calculate the square root of the right side: 7.158910532 Break this problem into two subproblems by setting (n + -4.5) equal to 7.158910532 and -7.158910532.Subproblem 1
n + -4.5 = 7.158910532 Simplifying n + -4.5 = 7.158910532 Reorder the terms: -4.5 + n = 7.158910532 Solving -4.5 + n = 7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = 7.158910532 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 7.158910532 + 4.5 n = 7.158910532 + 4.5 Combine like terms: 7.158910532 + 4.5 = 11.658910532 n = 11.658910532 Simplifying n = 11.658910532Subproblem 2
n + -4.5 = -7.158910532 Simplifying n + -4.5 = -7.158910532 Reorder the terms: -4.5 + n = -7.158910532 Solving -4.5 + n = -7.158910532 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + n = -7.158910532 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -7.158910532 + 4.5 n = -7.158910532 + 4.5 Combine like terms: -7.158910532 + 4.5 = -2.658910532 n = -2.658910532 Simplifying n = -2.658910532Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.658910532, -2.658910532}
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