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Simplifying (n + 1)(n + 2) = 452 Reorder the terms: (1 + n)(n + 2) = 452 Reorder the terms: (1 + n)(2 + n) = 452 Multiply (1 + n) * (2 + n) (1(2 + n) + n(2 + n)) = 452 ((2 * 1 + n * 1) + n(2 + n)) = 452 ((2 + 1n) + n(2 + n)) = 452 (2 + 1n + (2 * n + n * n)) = 452 (2 + 1n + (2n + n2)) = 452 Combine like terms: 1n + 2n = 3n (2 + 3n + n2) = 452 Solving 2 + 3n + n2 = 452 Solving for variable 'n'. Reorder the terms: 2 + -452 + 3n + n2 = 452 + -452 Combine like terms: 2 + -452 = -450 -450 + 3n + n2 = 452 + -452 Combine like terms: 452 + -452 = 0 -450 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '450' to each side of the equation. -450 + 3n + 450 + n2 = 0 + 450 Reorder the terms: -450 + 450 + 3n + n2 = 0 + 450 Combine like terms: -450 + 450 = 0 0 + 3n + n2 = 0 + 450 3n + n2 = 0 + 450 Combine like terms: 0 + 450 = 450 3n + n2 = 450 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 450 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 450 + 2.25 Combine like terms: 450 + 2.25 = 452.25 2.25 + 3n + n2 = 452.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 452.25 Calculate the square root of the right side: 21.266170318 Break this problem into two subproblems by setting (n + 1.5) equal to 21.266170318 and -21.266170318.Subproblem 1
n + 1.5 = 21.266170318 Simplifying n + 1.5 = 21.266170318 Reorder the terms: 1.5 + n = 21.266170318 Solving 1.5 + n = 21.266170318 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 21.266170318 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 21.266170318 + -1.5 n = 21.266170318 + -1.5 Combine like terms: 21.266170318 + -1.5 = 19.766170318 n = 19.766170318 Simplifying n = 19.766170318Subproblem 2
n + 1.5 = -21.266170318 Simplifying n + 1.5 = -21.266170318 Reorder the terms: 1.5 + n = -21.266170318 Solving 1.5 + n = -21.266170318 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -21.266170318 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -21.266170318 + -1.5 n = -21.266170318 + -1.5 Combine like terms: -21.266170318 + -1.5 = -22.766170318 n = -22.766170318 Simplifying n = -22.766170318Solution
The solution to the problem is based on the solutions from the subproblems. n = {19.766170318, -22.766170318}
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