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Simplifying n(n + 1) + (n + 2) = 156 Reorder the terms: n(1 + n) + (n + 2) = 156 (1 * n + n * n) + (n + 2) = 156 (1n + n2) + (n + 2) = 156 Reorder the terms: 1n + n2 + (2 + n) = 156 Remove parenthesis around (2 + n) 1n + n2 + 2 + n = 156 Reorder the terms: 2 + 1n + n + n2 = 156 Combine like terms: 1n + n = 2n 2 + 2n + n2 = 156 Solving 2 + 2n + n2 = 156 Solving for variable 'n'. Reorder the terms: 2 + -156 + 2n + n2 = 156 + -156 Combine like terms: 2 + -156 = -154 -154 + 2n + n2 = 156 + -156 Combine like terms: 156 + -156 = 0 -154 + 2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '154' to each side of the equation. -154 + 2n + 154 + n2 = 0 + 154 Reorder the terms: -154 + 154 + 2n + n2 = 0 + 154 Combine like terms: -154 + 154 = 0 0 + 2n + n2 = 0 + 154 2n + n2 = 0 + 154 Combine like terms: 0 + 154 = 154 2n + n2 = 154 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 = 154 + 1 Reorder the terms: 1 + 2n + n2 = 154 + 1 Combine like terms: 154 + 1 = 155 1 + 2n + n2 = 155 Factor a perfect square on the left side: (n + 1)(n + 1) = 155 Calculate the square root of the right side: 12.449899598 Break this problem into two subproblems by setting (n + 1) equal to 12.449899598 and -12.449899598.Subproblem 1
n + 1 = 12.449899598 Simplifying n + 1 = 12.449899598 Reorder the terms: 1 + n = 12.449899598 Solving 1 + n = 12.449899598 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 = 12.449899598 + -1 Combine like terms: 1 + -1 = 0 0 + n = 12.449899598 + -1 n = 12.449899598 + -1 Combine like terms: 12.449899598 + -1 = 11.449899598 n = 11.449899598 Simplifying n = 11.449899598Subproblem 2
n + 1 = -12.449899598 Simplifying n + 1 = -12.449899598 Reorder the terms: 1 + n = -12.449899598 Solving 1 + n = -12.449899598 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 = -12.449899598 + -1 Combine like terms: 1 + -1 = 0 0 + n = -12.449899598 + -1 n = -12.449899598 + -1 Combine like terms: -12.449899598 + -1 = -13.449899598 n = -13.449899598 Simplifying n = -13.449899598Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.449899598, -13.449899598}
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