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Simplifying n(n + 1) = 84 Reorder the terms: n(1 + n) = 84 (1 * n + n * n) = 84 (1n + n2) = 84 Solving 1n + n2 = 84 Solving for variable 'n'. Reorder the terms: -84 + 1n + n2 = 84 + -84 Combine like terms: 84 + -84 = 0 -84 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '84' to each side of the equation. -84 + 1n + 84 + n2 = 0 + 84 Reorder the terms: -84 + 84 + 1n + n2 = 0 + 84 Combine like terms: -84 + 84 = 0 0 + 1n + n2 = 0 + 84 1n + n2 = 0 + 84 Combine like terms: 0 + 84 = 84 1n + n2 = 84 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 = 84 + 0.25 Reorder the terms: 0.25 + n + n2 = 84 + 0.25 Combine like terms: 84 + 0.25 = 84.25 0.25 + n + n2 = 84.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 84.25 Calculate the square root of the right side: 9.178779875 Break this problem into two subproblems by setting (n + 0.5) equal to 9.178779875 and -9.178779875.Subproblem 1
n + 0.5 = 9.178779875 Simplifying n + 0.5 = 9.178779875 Reorder the terms: 0.5 + n = 9.178779875 Solving 0.5 + n = 9.178779875 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 = 9.178779875 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 9.178779875 + -0.5 n = 9.178779875 + -0.5 Combine like terms: 9.178779875 + -0.5 = 8.678779875 n = 8.678779875 Simplifying n = 8.678779875Subproblem 2
n + 0.5 = -9.178779875 Simplifying n + 0.5 = -9.178779875 Reorder the terms: 0.5 + n = -9.178779875 Solving 0.5 + n = -9.178779875 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 = -9.178779875 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -9.178779875 + -0.5 n = -9.178779875 + -0.5 Combine like terms: -9.178779875 + -0.5 = -9.678779875 n = -9.678779875 Simplifying n = -9.678779875Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.678779875, -9.678779875}
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