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Simplifying n(n + 1) = 224 Reorder the terms: n(1 + n) = 224 (1 * n + n * n) = 224 (1n + n2) = 224 Solving 1n + n2 = 224 Solving for variable 'n'. Reorder the terms: -224 + 1n + n2 = 224 + -224 Combine like terms: 224 + -224 = 0 -224 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '224' to each side of the equation. -224 + 1n + 224 + n2 = 0 + 224 Reorder the terms: -224 + 224 + 1n + n2 = 0 + 224 Combine like terms: -224 + 224 = 0 0 + 1n + n2 = 0 + 224 1n + n2 = 0 + 224 Combine like terms: 0 + 224 = 224 1n + n2 = 224 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 = 224 + 0.25 Reorder the terms: 0.25 + n + n2 = 224 + 0.25 Combine like terms: 224 + 0.25 = 224.25 0.25 + n + n2 = 224.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 224.25 Calculate the square root of the right side: 14.974979132 Break this problem into two subproblems by setting (n + 0.5) equal to 14.974979132 and -14.974979132.Subproblem 1
n + 0.5 = 14.974979132 Simplifying n + 0.5 = 14.974979132 Reorder the terms: 0.5 + n = 14.974979132 Solving 0.5 + n = 14.974979132 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 = 14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 14.974979132 + -0.5 n = 14.974979132 + -0.5 Combine like terms: 14.974979132 + -0.5 = 14.474979132 n = 14.474979132 Simplifying n = 14.474979132Subproblem 2
n + 0.5 = -14.974979132 Simplifying n + 0.5 = -14.974979132 Reorder the terms: 0.5 + n = -14.974979132 Solving 0.5 + n = -14.974979132 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 = -14.974979132 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -14.974979132 + -0.5 n = -14.974979132 + -0.5 Combine like terms: -14.974979132 + -0.5 = -15.474979132 n = -15.474979132 Simplifying n = -15.474979132Solution
The solution to the problem is based on the solutions from the subproblems. n = {14.474979132, -15.474979132}
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