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Simplifying 16n2 + 144n = 14 Reorder the terms: 144n + 16n2 = 14 Solving 144n + 16n2 = 14 Solving for variable 'n'. Reorder the terms: -14 + 144n + 16n2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 144n + 16n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 72n + 8n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 72n + 8n2)' equal to zero and attempt to solve: Simplifying -7 + 72n + 8n2 = 0 Solving -7 + 72n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + 9n + n2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 9n + 0.875 + n2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + 9n + n2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 9n + n2 = 0 + 0.875 9n + n2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 9n + n2 = 0.875 The n term is 9n. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9n + 20.25 + n2 = 0.875 + 20.25 Reorder the terms: 20.25 + 9n + n2 = 0.875 + 20.25 Combine like terms: 0.875 + 20.25 = 21.125 20.25 + 9n + n2 = 21.125 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 21.125 Calculate the square root of the right side: 4.596194078 Break this problem into two subproblems by setting (n + 4.5) equal to 4.596194078 and -4.596194078.Subproblem 1
n + 4.5 = 4.596194078 Simplifying n + 4.5 = 4.596194078 Reorder the terms: 4.5 + n = 4.596194078 Solving 4.5 + n = 4.596194078 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 4.596194078 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 4.596194078 + -4.5 n = 4.596194078 + -4.5 Combine like terms: 4.596194078 + -4.5 = 0.096194078 n = 0.096194078 Simplifying n = 0.096194078Subproblem 2
n + 4.5 = -4.596194078 Simplifying n + 4.5 = -4.596194078 Reorder the terms: 4.5 + n = -4.596194078 Solving 4.5 + n = -4.596194078 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -4.596194078 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -4.596194078 + -4.5 n = -4.596194078 + -4.5 Combine like terms: -4.596194078 + -4.5 = -9.096194078 n = -9.096194078 Simplifying n = -9.096194078Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.096194078, -9.096194078}Solution
n = {0.096194078, -9.096194078}
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