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Simplifying n2 + 13n + -30 = -7 Reorder the terms: -30 + 13n + n2 = -7 Solving -30 + 13n + n2 = -7 Solving for variable 'n'. Reorder the terms: -30 + 7 + 13n + n2 = -7 + 7 Combine like terms: -30 + 7 = -23 -23 + 13n + n2 = -7 + 7 Combine like terms: -7 + 7 = 0 -23 + 13n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 13n + 23 + n2 = 0 + 23 Reorder the terms: -23 + 23 + 13n + n2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 13n + n2 = 0 + 23 13n + n2 = 0 + 23 Combine like terms: 0 + 23 = 23 13n + n2 = 23 The n term is 13n. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13n + 42.25 + n2 = 23 + 42.25 Reorder the terms: 42.25 + 13n + n2 = 23 + 42.25 Combine like terms: 23 + 42.25 = 65.25 42.25 + 13n + n2 = 65.25 Factor a perfect square on the left side: (n + 6.5)(n + 6.5) = 65.25 Calculate the square root of the right side: 8.077747211 Break this problem into two subproblems by setting (n + 6.5) equal to 8.077747211 and -8.077747211.Subproblem 1
n + 6.5 = 8.077747211 Simplifying n + 6.5 = 8.077747211 Reorder the terms: 6.5 + n = 8.077747211 Solving 6.5 + n = 8.077747211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = 8.077747211 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = 8.077747211 + -6.5 n = 8.077747211 + -6.5 Combine like terms: 8.077747211 + -6.5 = 1.577747211 n = 1.577747211 Simplifying n = 1.577747211Subproblem 2
n + 6.5 = -8.077747211 Simplifying n + 6.5 = -8.077747211 Reorder the terms: 6.5 + n = -8.077747211 Solving 6.5 + n = -8.077747211 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + n = -8.077747211 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + n = -8.077747211 + -6.5 n = -8.077747211 + -6.5 Combine like terms: -8.077747211 + -6.5 = -14.577747211 n = -14.577747211 Simplifying n = -14.577747211Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.577747211, -14.577747211}
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