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Simplifying (n + 5)(n + -6) = 21 Reorder the terms: (5 + n)(n + -6) = 21 Reorder the terms: (5 + n)(-6 + n) = 21 Multiply (5 + n) * (-6 + n) (5(-6 + n) + n(-6 + n)) = 21 ((-6 * 5 + n * 5) + n(-6 + n)) = 21 ((-30 + 5n) + n(-6 + n)) = 21 (-30 + 5n + (-6 * n + n * n)) = 21 (-30 + 5n + (-6n + n2)) = 21 Combine like terms: 5n + -6n = -1n (-30 + -1n + n2) = 21 Solving -30 + -1n + n2 = 21 Solving for variable 'n'. Reorder the terms: -30 + -21 + -1n + n2 = 21 + -21 Combine like terms: -30 + -21 = -51 -51 + -1n + n2 = 21 + -21 Combine like terms: 21 + -21 = 0 -51 + -1n + n2 = 0 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. -1n + 0.25 + n2 = 51 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 51 + 0.25 Combine like terms: 51 + 0.25 = 51.25 0.25 + -1n + 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 = 7.658910532 n = 7.658910532 Simplifying n = 7.658910532Subproblem 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 = -6.658910532 n = -6.658910532 Simplifying n = -6.658910532Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.658910532, -6.658910532}
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