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Simplifying n(n + 1) = 2000 Reorder the terms: n(1 + n) = 2000 (1 * n + n * n) = 2000 (1n + n2) = 2000 Solving 1n + n2 = 2000 Solving for variable 'n'. Reorder the terms: -2000 + 1n + n2 = 2000 + -2000 Combine like terms: 2000 + -2000 = 0 -2000 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '2000' to each side of the equation. -2000 + 1n + 2000 + n2 = 0 + 2000 Reorder the terms: -2000 + 2000 + 1n + n2 = 0 + 2000 Combine like terms: -2000 + 2000 = 0 0 + 1n + n2 = 0 + 2000 1n + n2 = 0 + 2000 Combine like terms: 0 + 2000 = 2000 1n + n2 = 2000 The n term is 1n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 2000 + 0.25 Reorder the terms: 0.25 + n + n2 = 2000 + 0.25 Combine like terms: 2000 + 0.25 = 2000.25 0.25 + n + n2 = 2000.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 2000.25 Calculate the square root of the right side: 44.724154548 Break this problem into two subproblems by setting (n + 0.5) equal to 44.724154548 and -44.724154548.Subproblem 1
n + 0.5 = 44.724154548 Simplifying n + 0.5 = 44.724154548 Reorder the terms: 0.5 + n = 44.724154548 Solving 0.5 + n = 44.724154548 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 44.724154548 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 44.724154548 + -0.5 n = 44.724154548 + -0.5 Combine like terms: 44.724154548 + -0.5 = 44.224154548 n = 44.224154548 Simplifying n = 44.224154548Subproblem 2
n + 0.5 = -44.724154548 Simplifying n + 0.5 = -44.724154548 Reorder the terms: 0.5 + n = -44.724154548 Solving 0.5 + n = -44.724154548 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -44.724154548 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -44.724154548 + -0.5 n = -44.724154548 + -0.5 Combine like terms: -44.724154548 + -0.5 = -45.224154548 n = -45.224154548 Simplifying n = -45.224154548Solution
The solution to the problem is based on the solutions from the subproblems. n = {44.224154548, -45.224154548}
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