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Simplifying n(n + 1) = 440 Reorder the terms: n(1 + n) = 440 (1 * n + n * n) = 440 (1n + n2) = 440 Solving 1n + n2 = 440 Solving for variable 'n'. Reorder the terms: -440 + 1n + n2 = 440 + -440 Combine like terms: 440 + -440 = 0 -440 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '440' to each side of the equation. -440 + 1n + 440 + n2 = 0 + 440 Reorder the terms: -440 + 440 + 1n + n2 = 0 + 440 Combine like terms: -440 + 440 = 0 0 + 1n + n2 = 0 + 440 1n + n2 = 0 + 440 Combine like terms: 0 + 440 = 440 1n + n2 = 440 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 = 440 + 0.25 Reorder the terms: 0.25 + n + n2 = 440 + 0.25 Combine like terms: 440 + 0.25 = 440.25 0.25 + n + n2 = 440.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 440.25 Calculate the square root of the right side: 20.982135258 Break this problem into two subproblems by setting (n + 0.5) equal to 20.982135258 and -20.982135258.Subproblem 1
n + 0.5 = 20.982135258 Simplifying n + 0.5 = 20.982135258 Reorder the terms: 0.5 + n = 20.982135258 Solving 0.5 + n = 20.982135258 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 = 20.982135258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 20.982135258 + -0.5 n = 20.982135258 + -0.5 Combine like terms: 20.982135258 + -0.5 = 20.482135258 n = 20.482135258 Simplifying n = 20.482135258Subproblem 2
n + 0.5 = -20.982135258 Simplifying n + 0.5 = -20.982135258 Reorder the terms: 0.5 + n = -20.982135258 Solving 0.5 + n = -20.982135258 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 = -20.982135258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -20.982135258 + -0.5 n = -20.982135258 + -0.5 Combine like terms: -20.982135258 + -0.5 = -21.482135258 n = -21.482135258 Simplifying n = -21.482135258Solution
The solution to the problem is based on the solutions from the subproblems. n = {20.482135258, -21.482135258}
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