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Simplifying n(n + 1) + (n + 2) = 54 Reorder the terms: n(1 + n) + (n + 2) = 54 (1 * n + n * n) + (n + 2) = 54 (1n + n2) + (n + 2) = 54 Reorder the terms: 1n + n2 + (2 + n) = 54 Remove parenthesis around (2 + n) 1n + n2 + 2 + n = 54 Reorder the terms: 2 + 1n + n + n2 = 54 Combine like terms: 1n + n = 2n 2 + 2n + n2 = 54 Solving 2 + 2n + n2 = 54 Solving for variable 'n'. Reorder the terms: 2 + -54 + 2n + n2 = 54 + -54 Combine like terms: 2 + -54 = -52 -52 + 2n + n2 = 54 + -54 Combine like terms: 54 + -54 = 0 -52 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '52' to each side of the equation. -52 + 2n + 52 + n2 = 0 + 52 Reorder the terms: -52 + 52 + 2n + n2 = 0 + 52 Combine like terms: -52 + 52 = 0 0 + 2n + n2 = 0 + 52 2n + n2 = 0 + 52 Combine like terms: 0 + 52 = 52 2n + n2 = 52 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 = 52 + 1 Reorder the terms: 1 + 2n + n2 = 52 + 1 Combine like terms: 52 + 1 = 53 1 + 2n + n2 = 53 Factor a perfect square on the left side: (n + 1)(n + 1) = 53 Calculate the square root of the right side: 7.280109889 Break this problem into two subproblems by setting (n + 1) equal to 7.280109889 and -7.280109889.Subproblem 1
n + 1 = 7.280109889 Simplifying n + 1 = 7.280109889 Reorder the terms: 1 + n = 7.280109889 Solving 1 + n = 7.280109889 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 = 7.280109889 + -1 Combine like terms: 1 + -1 = 0 0 + n = 7.280109889 + -1 n = 7.280109889 + -1 Combine like terms: 7.280109889 + -1 = 6.280109889 n = 6.280109889 Simplifying n = 6.280109889Subproblem 2
n + 1 = -7.280109889 Simplifying n + 1 = -7.280109889 Reorder the terms: 1 + n = -7.280109889 Solving 1 + n = -7.280109889 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 = -7.280109889 + -1 Combine like terms: 1 + -1 = 0 0 + n = -7.280109889 + -1 n = -7.280109889 + -1 Combine like terms: -7.280109889 + -1 = -8.280109889 n = -8.280109889 Simplifying n = -8.280109889Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.280109889, -8.280109889}
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