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Simplifying 16n2 + -144n + 14 = 0 Reorder the terms: 14 + -144n + 16n2 = 0 Solving 14 + -144n + 16n2 = 0 Solving for variable 'n'. 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 = 19.375 20.25 + -9n + n2 = 19.375 Factor a perfect square on the left side: (n + -4.5)(n + -4.5) = 19.375 Calculate the square root of the right side: 4.401704215 Break this problem into two subproblems by setting (n + -4.5) equal to 4.401704215 and -4.401704215.Subproblem 1
n + -4.5 = 4.401704215 Simplifying n + -4.5 = 4.401704215 Reorder the terms: -4.5 + n = 4.401704215 Solving -4.5 + n = 4.401704215 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 = 4.401704215 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = 4.401704215 + 4.5 n = 4.401704215 + 4.5 Combine like terms: 4.401704215 + 4.5 = 8.901704215 n = 8.901704215 Simplifying n = 8.901704215Subproblem 2
n + -4.5 = -4.401704215 Simplifying n + -4.5 = -4.401704215 Reorder the terms: -4.5 + n = -4.401704215 Solving -4.5 + n = -4.401704215 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 = -4.401704215 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + n = -4.401704215 + 4.5 n = -4.401704215 + 4.5 Combine like terms: -4.401704215 + 4.5 = 0.098295785 n = 0.098295785 Simplifying n = 0.098295785Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.901704215, 0.098295785}Solution
n = {8.901704215, 0.098295785}
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