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Simplifying 0 = n(120 + -1n) + -50 0 = (120 * n + -1n * n) + -50 0 = (120n + -1n2) + -50 Reorder the terms: 0 = -50 + 120n + -1n2 Solving 0 = -50 + 120n + -1n2 Solving for variable 'n'. Combine like terms: 0 + 50 = 50 50 + -120n + n2 = -50 + 120n + -1n2 + 50 + -120n + n2 Reorder the terms: 50 + -120n + n2 = -50 + 50 + 120n + -120n + -1n2 + n2 Combine like terms: -50 + 50 = 0 50 + -120n + n2 = 0 + 120n + -120n + -1n2 + n2 50 + -120n + n2 = 120n + -120n + -1n2 + n2 Combine like terms: 120n + -120n = 0 50 + -120n + n2 = 0 + -1n2 + n2 50 + -120n + n2 = -1n2 + n2 Combine like terms: -1n2 + n2 = 0 50 + -120n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-50' to each side of the equation. 50 + -120n + -50 + n2 = 0 + -50 Reorder the terms: 50 + -50 + -120n + n2 = 0 + -50 Combine like terms: 50 + -50 = 0 0 + -120n + n2 = 0 + -50 -120n + n2 = 0 + -50 Combine like terms: 0 + -50 = -50 -120n + n2 = -50 The n term is -120n. Take half its coefficient (-60). Square it (3600) and add it to both sides. Add '3600' to each side of the equation. -120n + 3600 + n2 = -50 + 3600 Reorder the terms: 3600 + -120n + n2 = -50 + 3600 Combine like terms: -50 + 3600 = 3550 3600 + -120n + n2 = 3550 Factor a perfect square on the left side: (n + -60)(n + -60) = 3550 Calculate the square root of the right side: 59.581876439 Break this problem into two subproblems by setting (n + -60) equal to 59.581876439 and -59.581876439.Subproblem 1
n + -60 = 59.581876439 Simplifying n + -60 = 59.581876439 Reorder the terms: -60 + n = 59.581876439 Solving -60 + n = 59.581876439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + n = 59.581876439 + 60 Combine like terms: -60 + 60 = 0 0 + n = 59.581876439 + 60 n = 59.581876439 + 60 Combine like terms: 59.581876439 + 60 = 119.581876439 n = 119.581876439 Simplifying n = 119.581876439Subproblem 2
n + -60 = -59.581876439 Simplifying n + -60 = -59.581876439 Reorder the terms: -60 + n = -59.581876439 Solving -60 + n = -59.581876439 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '60' to each side of the equation. -60 + 60 + n = -59.581876439 + 60 Combine like terms: -60 + 60 = 0 0 + n = -59.581876439 + 60 n = -59.581876439 + 60 Combine like terms: -59.581876439 + 60 = 0.418123561 n = 0.418123561 Simplifying n = 0.418123561Solution
The solution to the problem is based on the solutions from the subproblems. n = {119.581876439, 0.418123561}
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