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Simplifying n2 + -3n + -18 = 45 Reorder the terms: -18 + -3n + n2 = 45 Solving -18 + -3n + n2 = 45 Solving for variable 'n'. Reorder the terms: -18 + -45 + -3n + n2 = 45 + -45 Combine like terms: -18 + -45 = -63 -63 + -3n + n2 = 45 + -45 Combine like terms: 45 + -45 = 0 -63 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '63' to each side of the equation. -63 + -3n + 63 + n2 = 0 + 63 Reorder the terms: -63 + 63 + -3n + n2 = 0 + 63 Combine like terms: -63 + 63 = 0 0 + -3n + n2 = 0 + 63 -3n + n2 = 0 + 63 Combine like terms: 0 + 63 = 63 -3n + n2 = 63 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 = 63 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 63 + 2.25 Combine like terms: 63 + 2.25 = 65.25 2.25 + -3n + n2 = 65.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 65.25 Calculate the square root of the right side: 8.077747211 Break this problem into two subproblems by setting (n + -1.5) equal to 8.077747211 and -8.077747211.Subproblem 1
n + -1.5 = 8.077747211 Simplifying n + -1.5 = 8.077747211 Reorder the terms: -1.5 + n = 8.077747211 Solving -1.5 + n = 8.077747211 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 = 8.077747211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 8.077747211 + 1.5 n = 8.077747211 + 1.5 Combine like terms: 8.077747211 + 1.5 = 9.577747211 n = 9.577747211 Simplifying n = 9.577747211Subproblem 2
n + -1.5 = -8.077747211 Simplifying n + -1.5 = -8.077747211 Reorder the terms: -1.5 + n = -8.077747211 Solving -1.5 + n = -8.077747211 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 = -8.077747211 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -8.077747211 + 1.5 n = -8.077747211 + 1.5 Combine like terms: -8.077747211 + 1.5 = -6.577747211 n = -6.577747211 Simplifying n = -6.577747211Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.577747211, -6.577747211}
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