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Simplifying (2n)(11n + -14) = 180 Remove parenthesis around (2n) 2n(11n + -14) = 180 Reorder the terms: 2n(-14 + 11n) = 180 (-14 * 2n + 11n * 2n) = 180 (-28n + 22n2) = 180 Solving -28n + 22n2 = 180 Solving for variable 'n'. Reorder the terms: -180 + -28n + 22n2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + -28n + 22n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-90 + -14n + 11n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-90 + -14n + 11n2)' equal to zero and attempt to solve: Simplifying -90 + -14n + 11n2 = 0 Solving -90 + -14n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -8.181818182 + -1.272727273n + n2 = 0 Move the constant term to the right: Add '8.181818182' to each side of the equation. -8.181818182 + -1.272727273n + 8.181818182 + n2 = 0 + 8.181818182 Reorder the terms: -8.181818182 + 8.181818182 + -1.272727273n + n2 = 0 + 8.181818182 Combine like terms: -8.181818182 + 8.181818182 = 0.000000000 0.000000000 + -1.272727273n + n2 = 0 + 8.181818182 -1.272727273n + n2 = 0 + 8.181818182 Combine like terms: 0 + 8.181818182 = 8.181818182 -1.272727273n + n2 = 8.181818182 The n term is -1.272727273n. Take half its coefficient (-0.6363636365). Square it (0.4049586779) and add it to both sides. Add '0.4049586779' to each side of the equation. -1.272727273n + 0.4049586779 + n2 = 8.181818182 + 0.4049586779 Reorder the terms: 0.4049586779 + -1.272727273n + n2 = 8.181818182 + 0.4049586779 Combine like terms: 8.181818182 + 0.4049586779 = 8.5867768599 0.4049586779 + -1.272727273n + n2 = 8.5867768599 Factor a perfect square on the left side: (n + -0.6363636365)(n + -0.6363636365) = 8.5867768599 Calculate the square root of the right side: 2.930320266 Break this problem into two subproblems by setting (n + -0.6363636365) equal to 2.930320266 and -2.930320266.Subproblem 1
n + -0.6363636365 = 2.930320266 Simplifying n + -0.6363636365 = 2.930320266 Reorder the terms: -0.6363636365 + n = 2.930320266 Solving -0.6363636365 + n = 2.930320266 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6363636365' to each side of the equation. -0.6363636365 + 0.6363636365 + n = 2.930320266 + 0.6363636365 Combine like terms: -0.6363636365 + 0.6363636365 = 0.0000000000 0.0000000000 + n = 2.930320266 + 0.6363636365 n = 2.930320266 + 0.6363636365 Combine like terms: 2.930320266 + 0.6363636365 = 3.5666839025 n = 3.5666839025 Simplifying n = 3.5666839025Subproblem 2
n + -0.6363636365 = -2.930320266 Simplifying n + -0.6363636365 = -2.930320266 Reorder the terms: -0.6363636365 + n = -2.930320266 Solving -0.6363636365 + n = -2.930320266 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6363636365' to each side of the equation. -0.6363636365 + 0.6363636365 + n = -2.930320266 + 0.6363636365 Combine like terms: -0.6363636365 + 0.6363636365 = 0.0000000000 0.0000000000 + n = -2.930320266 + 0.6363636365 n = -2.930320266 + 0.6363636365 Combine like terms: -2.930320266 + 0.6363636365 = -2.2939566295 n = -2.2939566295 Simplifying n = -2.2939566295Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.5666839025, -2.2939566295}Solution
n = {3.5666839025, -2.2939566295}
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