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Simplifying 3n + 18 + n2 = 120 Reorder the terms: 18 + 3n + n2 = 120 Solving 18 + 3n + n2 = 120 Solving for variable 'n'. Reorder the terms: 18 + -120 + 3n + n2 = 120 + -120 Combine like terms: 18 + -120 = -102 -102 + 3n + n2 = 120 + -120 Combine like terms: 120 + -120 = 0 -102 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '102' to each side of the equation. -102 + 3n + 102 + n2 = 0 + 102 Reorder the terms: -102 + 102 + 3n + n2 = 0 + 102 Combine like terms: -102 + 102 = 0 0 + 3n + n2 = 0 + 102 3n + n2 = 0 + 102 Combine like terms: 0 + 102 = 102 3n + n2 = 102 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 102 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 102 + 2.25 Combine like terms: 102 + 2.25 = 104.25 2.25 + 3n + n2 = 104.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 104.25 Calculate the square root of the right side: 10.210288928 Break this problem into two subproblems by setting (n + 1.5) equal to 10.210288928 and -10.210288928.Subproblem 1
n + 1.5 = 10.210288928 Simplifying n + 1.5 = 10.210288928 Reorder the terms: 1.5 + n = 10.210288928 Solving 1.5 + n = 10.210288928 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 10.210288928 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 10.210288928 + -1.5 n = 10.210288928 + -1.5 Combine like terms: 10.210288928 + -1.5 = 8.710288928 n = 8.710288928 Simplifying n = 8.710288928Subproblem 2
n + 1.5 = -10.210288928 Simplifying n + 1.5 = -10.210288928 Reorder the terms: 1.5 + n = -10.210288928 Solving 1.5 + n = -10.210288928 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -10.210288928 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -10.210288928 + -1.5 n = -10.210288928 + -1.5 Combine like terms: -10.210288928 + -1.5 = -11.710288928 n = -11.710288928 Simplifying n = -11.710288928Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.710288928, -11.710288928}
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