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Simplifying n2 + 2n = 85 Reorder the terms: 2n + n2 = 85 Solving 2n + n2 = 85 Solving for variable 'n'. Reorder the terms: -85 + 2n + n2 = 85 + -85 Combine like terms: 85 + -85 = 0 -85 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '85' to each side of the equation. -85 + 2n + 85 + n2 = 0 + 85 Reorder the terms: -85 + 85 + 2n + n2 = 0 + 85 Combine like terms: -85 + 85 = 0 0 + 2n + n2 = 0 + 85 2n + n2 = 0 + 85 Combine like terms: 0 + 85 = 85 2n + n2 = 85 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 = 85 + 1 Reorder the terms: 1 + 2n + n2 = 85 + 1 Combine like terms: 85 + 1 = 86 1 + 2n + n2 = 86 Factor a perfect square on the left side: (n + 1)(n + 1) = 86 Calculate the square root of the right side: 9.273618495 Break this problem into two subproblems by setting (n + 1) equal to 9.273618495 and -9.273618495.Subproblem 1
n + 1 = 9.273618495 Simplifying n + 1 = 9.273618495 Reorder the terms: 1 + n = 9.273618495 Solving 1 + n = 9.273618495 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 = 9.273618495 + -1 Combine like terms: 1 + -1 = 0 0 + n = 9.273618495 + -1 n = 9.273618495 + -1 Combine like terms: 9.273618495 + -1 = 8.273618495 n = 8.273618495 Simplifying n = 8.273618495Subproblem 2
n + 1 = -9.273618495 Simplifying n + 1 = -9.273618495 Reorder the terms: 1 + n = -9.273618495 Solving 1 + n = -9.273618495 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 = -9.273618495 + -1 Combine like terms: 1 + -1 = 0 0 + n = -9.273618495 + -1 n = -9.273618495 + -1 Combine like terms: -9.273618495 + -1 = -10.273618495 n = -10.273618495 Simplifying n = -10.273618495Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.273618495, -10.273618495}
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