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Simplifying 3n3 + -9n2 + 3n = 0 Reorder the terms: 3n + -9n2 + 3n3 = 0 Solving 3n + -9n2 + 3n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3n'. 3n(1 + -3n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(1 + -3n + n2)' equal to zero and attempt to solve: Simplifying 1 + -3n + n2 = 0 Solving 1 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + -3n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3n + n2 = 0 + -1 -3n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3n + n2 = -1 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = -1 + 2.25 Reorder the terms: 2.25 + -3n + n2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3n + n2 = 1.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (n + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
n + -1.5 = 1.118033989 Simplifying n + -1.5 = 1.118033989 Reorder the terms: -1.5 + n = 1.118033989 Solving -1.5 + n = 1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 1.118033989 + 1.5 n = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 n = 2.618033989 Simplifying n = 2.618033989Subproblem 2
n + -1.5 = -1.118033989 Simplifying n + -1.5 = -1.118033989 Reorder the terms: -1.5 + n = -1.118033989 Solving -1.5 + n = -1.118033989 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -1.118033989 + 1.5 n = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 n = 0.381966011 Simplifying n = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.618033989, 0.381966011}Solution
n = {0, 2.618033989, 0.381966011}
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