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Simplifying n(n + 1) = 126 Reorder the terms: n(1 + n) = 126 (1 * n + n * n) = 126 (1n + n2) = 126 Solving 1n + n2 = 126 Solving for variable 'n'. Reorder the terms: -126 + 1n + n2 = 126 + -126 Combine like terms: 126 + -126 = 0 -126 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '126' to each side of the equation. -126 + 1n + 126 + n2 = 0 + 126 Reorder the terms: -126 + 126 + 1n + n2 = 0 + 126 Combine like terms: -126 + 126 = 0 0 + 1n + n2 = 0 + 126 1n + n2 = 0 + 126 Combine like terms: 0 + 126 = 126 1n + n2 = 126 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 = 126 + 0.25 Reorder the terms: 0.25 + n + n2 = 126 + 0.25 Combine like terms: 126 + 0.25 = 126.25 0.25 + n + n2 = 126.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 126.25 Calculate the square root of the right side: 11.236102527 Break this problem into two subproblems by setting (n + 0.5) equal to 11.236102527 and -11.236102527.Subproblem 1
n + 0.5 = 11.236102527 Simplifying n + 0.5 = 11.236102527 Reorder the terms: 0.5 + n = 11.236102527 Solving 0.5 + n = 11.236102527 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 = 11.236102527 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 11.236102527 + -0.5 n = 11.236102527 + -0.5 Combine like terms: 11.236102527 + -0.5 = 10.736102527 n = 10.736102527 Simplifying n = 10.736102527Subproblem 2
n + 0.5 = -11.236102527 Simplifying n + 0.5 = -11.236102527 Reorder the terms: 0.5 + n = -11.236102527 Solving 0.5 + n = -11.236102527 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 = -11.236102527 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -11.236102527 + -0.5 n = -11.236102527 + -0.5 Combine like terms: -11.236102527 + -0.5 = -11.736102527 n = -11.736102527 Simplifying n = -11.736102527Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.736102527, -11.736102527}
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