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Simplifying n2 + 14n + -3 = 4n + -8 Reorder the terms: -3 + 14n + n2 = 4n + -8 Reorder the terms: -3 + 14n + n2 = -8 + 4n Solving -3 + 14n + n2 = -8 + 4n Solving for variable 'n'. Reorder the terms: -3 + 8 + 14n + -4n + n2 = -8 + 4n + 8 + -4n Combine like terms: -3 + 8 = 5 5 + 14n + -4n + n2 = -8 + 4n + 8 + -4n Combine like terms: 14n + -4n = 10n 5 + 10n + n2 = -8 + 4n + 8 + -4n Reorder the terms: 5 + 10n + n2 = -8 + 8 + 4n + -4n Combine like terms: -8 + 8 = 0 5 + 10n + n2 = 0 + 4n + -4n 5 + 10n + n2 = 4n + -4n Combine like terms: 4n + -4n = 0 5 + 10n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 10n + -5 + n2 = 0 + -5 Reorder the terms: 5 + -5 + 10n + n2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 10n + n2 = 0 + -5 10n + n2 = 0 + -5 Combine like terms: 0 + -5 = -5 10n + n2 = -5 The n term is 10n. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10n + 25 + n2 = -5 + 25 Reorder the terms: 25 + 10n + n2 = -5 + 25 Combine like terms: -5 + 25 = 20 25 + 10n + n2 = 20 Factor a perfect square on the left side: (n + 5)(n + 5) = 20 Calculate the square root of the right side: 4.472135955 Break this problem into two subproblems by setting (n + 5) equal to 4.472135955 and -4.472135955.Subproblem 1
n + 5 = 4.472135955 Simplifying n + 5 = 4.472135955 Reorder the terms: 5 + n = 4.472135955 Solving 5 + n = 4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = 4.472135955 + -5 Combine like terms: 5 + -5 = 0 0 + n = 4.472135955 + -5 n = 4.472135955 + -5 Combine like terms: 4.472135955 + -5 = -0.527864045 n = -0.527864045 Simplifying n = -0.527864045Subproblem 2
n + 5 = -4.472135955 Simplifying n + 5 = -4.472135955 Reorder the terms: 5 + n = -4.472135955 Solving 5 + n = -4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + n = -4.472135955 + -5 Combine like terms: 5 + -5 = 0 0 + n = -4.472135955 + -5 n = -4.472135955 + -5 Combine like terms: -4.472135955 + -5 = -9.472135955 n = -9.472135955 Simplifying n = -9.472135955Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.527864045, -9.472135955}
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