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Simplifying n2 + 2n + -85 = 5 Reorder the terms: -85 + 2n + n2 = 5 Solving -85 + 2n + n2 = 5 Solving for variable 'n'. Reorder the terms: -85 + -5 + 2n + n2 = 5 + -5 Combine like terms: -85 + -5 = -90 -90 + 2n + n2 = 5 + -5 Combine like terms: 5 + -5 = 0 -90 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '90' to each side of the equation. -90 + 2n + 90 + n2 = 0 + 90 Reorder the terms: -90 + 90 + 2n + n2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 2n + n2 = 0 + 90 2n + n2 = 0 + 90 Combine like terms: 0 + 90 = 90 2n + n2 = 90 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 90 + 1 Reorder the terms: 1 + 2n + n2 = 90 + 1 Combine like terms: 90 + 1 = 91 1 + 2n + n2 = 91 Factor a perfect square on the left side: (n + 1)(n + 1) = 91 Calculate the square root of the right side: 9.539392014 Break this problem into two subproblems by setting (n + 1) equal to 9.539392014 and -9.539392014.Subproblem 1
n + 1 = 9.539392014 Simplifying n + 1 = 9.539392014 Reorder the terms: 1 + n = 9.539392014 Solving 1 + n = 9.539392014 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 9.539392014 + -1 Combine like terms: 1 + -1 = 0 0 + n = 9.539392014 + -1 n = 9.539392014 + -1 Combine like terms: 9.539392014 + -1 = 8.539392014 n = 8.539392014 Simplifying n = 8.539392014Subproblem 2
n + 1 = -9.539392014 Simplifying n + 1 = -9.539392014 Reorder the terms: 1 + n = -9.539392014 Solving 1 + n = -9.539392014 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -9.539392014 + -1 Combine like terms: 1 + -1 = 0 0 + n = -9.539392014 + -1 n = -9.539392014 + -1 Combine like terms: -9.539392014 + -1 = -10.539392014 n = -10.539392014 Simplifying n = -10.539392014Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.539392014, -10.539392014}
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