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Simplifying n(n + -1) = 28 Reorder the terms: n(-1 + n) = 28 (-1 * n + n * n) = 28 (-1n + n2) = 28 Solving -1n + n2 = 28 Solving for variable 'n'. Reorder the terms: -28 + -1n + n2 = 28 + -28 Combine like terms: 28 + -28 = 0 -28 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '28' to each side of the equation. -28 + -1n + 28 + n2 = 0 + 28 Reorder the terms: -28 + 28 + -1n + n2 = 0 + 28 Combine like terms: -28 + 28 = 0 0 + -1n + n2 = 0 + 28 -1n + n2 = 0 + 28 Combine like terms: 0 + 28 = 28 -1n + n2 = 28 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. -1n + 0.25 + n2 = 28 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 28 + 0.25 Combine like terms: 28 + 0.25 = 28.25 0.25 + -1n + n2 = 28.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 28.25 Calculate the square root of the right side: 5.315072906 Break this problem into two subproblems by setting (n + -0.5) equal to 5.315072906 and -5.315072906.Subproblem 1
n + -0.5 = 5.315072906 Simplifying n + -0.5 = 5.315072906 Reorder the terms: -0.5 + n = 5.315072906 Solving -0.5 + n = 5.315072906 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 = 5.315072906 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 5.315072906 + 0.5 n = 5.315072906 + 0.5 Combine like terms: 5.315072906 + 0.5 = 5.815072906 n = 5.815072906 Simplifying n = 5.815072906Subproblem 2
n + -0.5 = -5.315072906 Simplifying n + -0.5 = -5.315072906 Reorder the terms: -0.5 + n = -5.315072906 Solving -0.5 + n = -5.315072906 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 = -5.315072906 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -5.315072906 + 0.5 n = -5.315072906 + 0.5 Combine like terms: -5.315072906 + 0.5 = -4.815072906 n = -4.815072906 Simplifying n = -4.815072906Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.815072906, -4.815072906}
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