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Simplifying n2 + 17n + -140 = 2n + -4n Reorder the terms: -140 + 17n + n2 = 2n + -4n Combine like terms: 2n + -4n = -2n -140 + 17n + n2 = -2n Solving -140 + 17n + n2 = -2n Solving for variable 'n'. Reorder the terms: -140 + 17n + 2n + n2 = -2n + 2n Combine like terms: 17n + 2n = 19n -140 + 19n + n2 = -2n + 2n Combine like terms: -2n + 2n = 0 -140 + 19n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '140' to each side of the equation. -140 + 19n + 140 + n2 = 0 + 140 Reorder the terms: -140 + 140 + 19n + n2 = 0 + 140 Combine like terms: -140 + 140 = 0 0 + 19n + n2 = 0 + 140 19n + n2 = 0 + 140 Combine like terms: 0 + 140 = 140 19n + n2 = 140 The n term is 19n. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19n + 90.25 + n2 = 140 + 90.25 Reorder the terms: 90.25 + 19n + n2 = 140 + 90.25 Combine like terms: 140 + 90.25 = 230.25 90.25 + 19n + n2 = 230.25 Factor a perfect square on the left side: (n + 9.5)(n + 9.5) = 230.25 Calculate the square root of the right side: 15.173990905 Break this problem into two subproblems by setting (n + 9.5) equal to 15.173990905 and -15.173990905.Subproblem 1
n + 9.5 = 15.173990905 Simplifying n + 9.5 = 15.173990905 Reorder the terms: 9.5 + n = 15.173990905 Solving 9.5 + n = 15.173990905 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = 15.173990905 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = 15.173990905 + -9.5 n = 15.173990905 + -9.5 Combine like terms: 15.173990905 + -9.5 = 5.673990905 n = 5.673990905 Simplifying n = 5.673990905Subproblem 2
n + 9.5 = -15.173990905 Simplifying n + 9.5 = -15.173990905 Reorder the terms: 9.5 + n = -15.173990905 Solving 9.5 + n = -15.173990905 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = -15.173990905 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = -15.173990905 + -9.5 n = -15.173990905 + -9.5 Combine like terms: -15.173990905 + -9.5 = -24.673990905 n = -24.673990905 Simplifying n = -24.673990905Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.673990905, -24.673990905}
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