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Simplifying n2 + -23n = 60 Reorder the terms: -23n + n2 = 60 Solving -23n + n2 = 60 Solving for variable 'n'. Reorder the terms: -60 + -23n + n2 = 60 + -60 Combine like terms: 60 + -60 = 0 -60 + -23n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '60' to each side of the equation. -60 + -23n + 60 + n2 = 0 + 60 Reorder the terms: -60 + 60 + -23n + n2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + -23n + n2 = 0 + 60 -23n + n2 = 0 + 60 Combine like terms: 0 + 60 = 60 -23n + n2 = 60 The n term is -23n. Take half its coefficient (-11.5). Square it (132.25) and add it to both sides. Add '132.25' to each side of the equation. -23n + 132.25 + n2 = 60 + 132.25 Reorder the terms: 132.25 + -23n + n2 = 60 + 132.25 Combine like terms: 60 + 132.25 = 192.25 132.25 + -23n + n2 = 192.25 Factor a perfect square on the left side: (n + -11.5)(n + -11.5) = 192.25 Calculate the square root of the right side: 13.865424624 Break this problem into two subproblems by setting (n + -11.5) equal to 13.865424624 and -13.865424624.Subproblem 1
n + -11.5 = 13.865424624 Simplifying n + -11.5 = 13.865424624 Reorder the terms: -11.5 + n = 13.865424624 Solving -11.5 + n = 13.865424624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + n = 13.865424624 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + n = 13.865424624 + 11.5 n = 13.865424624 + 11.5 Combine like terms: 13.865424624 + 11.5 = 25.365424624 n = 25.365424624 Simplifying n = 25.365424624Subproblem 2
n + -11.5 = -13.865424624 Simplifying n + -11.5 = -13.865424624 Reorder the terms: -11.5 + n = -13.865424624 Solving -11.5 + n = -13.865424624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.5' to each side of the equation. -11.5 + 11.5 + n = -13.865424624 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + n = -13.865424624 + 11.5 n = -13.865424624 + 11.5 Combine like terms: -13.865424624 + 11.5 = -2.365424624 n = -2.365424624 Simplifying n = -2.365424624Solution
The solution to the problem is based on the solutions from the subproblems. n = {25.365424624, -2.365424624}
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