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Simplifying n2 + -5n + 6n + -51 = 0 Reorder the terms: -51 + -5n + 6n + n2 = 0 Combine like terms: -5n + 6n = 1n -51 + 1n + n2 = 0 Solving -51 + 1n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '51' to each side of the equation. -51 + 1n + 51 + n2 = 0 + 51 Reorder the terms: -51 + 51 + 1n + n2 = 0 + 51 Combine like terms: -51 + 51 = 0 0 + 1n + n2 = 0 + 51 1n + n2 = 0 + 51 Combine like terms: 0 + 51 = 51 1n + n2 = 51 The n term is 1n. 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 = 51 + 0.25 Reorder the terms: 0.25 + n + n2 = 51 + 0.25 Combine like terms: 51 + 0.25 = 51.25 0.25 + n + n2 = 51.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 51.25 Calculate the square root of the right side: 7.158910532 Break this problem into two subproblems by setting (n + 0.5) equal to 7.158910532 and -7.158910532.Subproblem 1
n + 0.5 = 7.158910532 Simplifying n + 0.5 = 7.158910532 Reorder the terms: 0.5 + n = 7.158910532 Solving 0.5 + n = 7.158910532 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 = 7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 7.158910532 + -0.5 n = 7.158910532 + -0.5 Combine like terms: 7.158910532 + -0.5 = 6.658910532 n = 6.658910532 Simplifying n = 6.658910532Subproblem 2
n + 0.5 = -7.158910532 Simplifying n + 0.5 = -7.158910532 Reorder the terms: 0.5 + n = -7.158910532 Solving 0.5 + n = -7.158910532 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 = -7.158910532 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -7.158910532 + -0.5 n = -7.158910532 + -0.5 Combine like terms: -7.158910532 + -0.5 = -7.658910532 n = -7.658910532 Simplifying n = -7.658910532Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.658910532, -7.658910532}
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