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Simplifying n(n + 1) = 4012 Reorder the terms: n(1 + n) = 4012 (1 * n + n * n) = 4012 (1n + n2) = 4012 Solving 1n + n2 = 4012 Solving for variable 'n'. Reorder the terms: -4012 + 1n + n2 = 4012 + -4012 Combine like terms: 4012 + -4012 = 0 -4012 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '4012' to each side of the equation. -4012 + 1n + 4012 + n2 = 0 + 4012 Reorder the terms: -4012 + 4012 + 1n + n2 = 0 + 4012 Combine like terms: -4012 + 4012 = 0 0 + 1n + n2 = 0 + 4012 1n + n2 = 0 + 4012 Combine like terms: 0 + 4012 = 4012 1n + n2 = 4012 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 = 4012 + 0.25 Reorder the terms: 0.25 + n + n2 = 4012 + 0.25 Combine like terms: 4012 + 0.25 = 4012.25 0.25 + n + n2 = 4012.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 4012.25 Calculate the square root of the right side: 63.342323923 Break this problem into two subproblems by setting (n + 0.5) equal to 63.342323923 and -63.342323923.Subproblem 1
n + 0.5 = 63.342323923 Simplifying n + 0.5 = 63.342323923 Reorder the terms: 0.5 + n = 63.342323923 Solving 0.5 + n = 63.342323923 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 = 63.342323923 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 63.342323923 + -0.5 n = 63.342323923 + -0.5 Combine like terms: 63.342323923 + -0.5 = 62.842323923 n = 62.842323923 Simplifying n = 62.842323923Subproblem 2
n + 0.5 = -63.342323923 Simplifying n + 0.5 = -63.342323923 Reorder the terms: 0.5 + n = -63.342323923 Solving 0.5 + n = -63.342323923 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 = -63.342323923 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -63.342323923 + -0.5 n = -63.342323923 + -0.5 Combine like terms: -63.342323923 + -0.5 = -63.842323923 n = -63.842323923 Simplifying n = -63.842323923Solution
The solution to the problem is based on the solutions from the subproblems. n = {62.842323923, -63.842323923}
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