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Simplifying n(n + 1) + (n + 2) = 258 Reorder the terms: n(1 + n) + (n + 2) = 258 (1 * n + n * n) + (n + 2) = 258 (1n + n2) + (n + 2) = 258 Reorder the terms: 1n + n2 + (2 + n) = 258 Remove parenthesis around (2 + n) 1n + n2 + 2 + n = 258 Reorder the terms: 2 + 1n + n + n2 = 258 Combine like terms: 1n + n = 2n 2 + 2n + n2 = 258 Solving 2 + 2n + n2 = 258 Solving for variable 'n'. Reorder the terms: 2 + -258 + 2n + n2 = 258 + -258 Combine like terms: 2 + -258 = -256 -256 + 2n + n2 = 258 + -258 Combine like terms: 258 + -258 = 0 -256 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '256' to each side of the equation. -256 + 2n + 256 + n2 = 0 + 256 Reorder the terms: -256 + 256 + 2n + n2 = 0 + 256 Combine like terms: -256 + 256 = 0 0 + 2n + n2 = 0 + 256 2n + n2 = 0 + 256 Combine like terms: 0 + 256 = 256 2n + n2 = 256 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 = 256 + 1 Reorder the terms: 1 + 2n + n2 = 256 + 1 Combine like terms: 256 + 1 = 257 1 + 2n + n2 = 257 Factor a perfect square on the left side: (n + 1)(n + 1) = 257 Calculate the square root of the right side: 16.031219542 Break this problem into two subproblems by setting (n + 1) equal to 16.031219542 and -16.031219542.Subproblem 1
n + 1 = 16.031219542 Simplifying n + 1 = 16.031219542 Reorder the terms: 1 + n = 16.031219542 Solving 1 + n = 16.031219542 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 = 16.031219542 + -1 Combine like terms: 1 + -1 = 0 0 + n = 16.031219542 + -1 n = 16.031219542 + -1 Combine like terms: 16.031219542 + -1 = 15.031219542 n = 15.031219542 Simplifying n = 15.031219542Subproblem 2
n + 1 = -16.031219542 Simplifying n + 1 = -16.031219542 Reorder the terms: 1 + n = -16.031219542 Solving 1 + n = -16.031219542 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 = -16.031219542 + -1 Combine like terms: 1 + -1 = 0 0 + n = -16.031219542 + -1 n = -16.031219542 + -1 Combine like terms: -16.031219542 + -1 = -17.031219542 n = -17.031219542 Simplifying n = -17.031219542Solution
The solution to the problem is based on the solutions from the subproblems. n = {15.031219542, -17.031219542}
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