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Simplifying n(n + -1) = 226 Reorder the terms: n(-1 + n) = 226 (-1 * n + n * n) = 226 (-1n + n2) = 226 Solving -1n + n2 = 226 Solving for variable 'n'. Reorder the terms: -226 + -1n + n2 = 226 + -226 Combine like terms: 226 + -226 = 0 -226 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '226' to each side of the equation. -226 + -1n + 226 + n2 = 0 + 226 Reorder the terms: -226 + 226 + -1n + n2 = 0 + 226 Combine like terms: -226 + 226 = 0 0 + -1n + n2 = 0 + 226 -1n + n2 = 0 + 226 Combine like terms: 0 + 226 = 226 -1n + n2 = 226 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 = 226 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 226 + 0.25 Combine like terms: 226 + 0.25 = 226.25 0.25 + -1n + n2 = 226.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 226.25 Calculate the square root of the right side: 15.041608956 Break this problem into two subproblems by setting (n + -0.5) equal to 15.041608956 and -15.041608956.Subproblem 1
n + -0.5 = 15.041608956 Simplifying n + -0.5 = 15.041608956 Reorder the terms: -0.5 + n = 15.041608956 Solving -0.5 + n = 15.041608956 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 = 15.041608956 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 15.041608956 + 0.5 n = 15.041608956 + 0.5 Combine like terms: 15.041608956 + 0.5 = 15.541608956 n = 15.541608956 Simplifying n = 15.541608956Subproblem 2
n + -0.5 = -15.041608956 Simplifying n + -0.5 = -15.041608956 Reorder the terms: -0.5 + n = -15.041608956 Solving -0.5 + n = -15.041608956 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 = -15.041608956 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -15.041608956 + 0.5 n = -15.041608956 + 0.5 Combine like terms: -15.041608956 + 0.5 = -14.541608956 n = -14.541608956 Simplifying n = -14.541608956Solution
The solution to the problem is based on the solutions from the subproblems. n = {15.541608956, -14.541608956}
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