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Simplifying 9n3 + -27n2 + -3204n = 0 Reorder the terms: -3204n + -27n2 + 9n3 = 0 Solving -3204n + -27n2 + 9n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '9n'. 9n(-356 + -3n + n2) = 0 Ignore the factor 9.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 '(-356 + -3n + n2)' equal to zero and attempt to solve: Simplifying -356 + -3n + n2 = 0 Solving -356 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '356' to each side of the equation. -356 + -3n + 356 + n2 = 0 + 356 Reorder the terms: -356 + 356 + -3n + n2 = 0 + 356 Combine like terms: -356 + 356 = 0 0 + -3n + n2 = 0 + 356 -3n + n2 = 0 + 356 Combine like terms: 0 + 356 = 356 -3n + n2 = 356 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 = 356 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 356 + 2.25 Combine like terms: 356 + 2.25 = 358.25 2.25 + -3n + n2 = 358.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 358.25 Calculate the square root of the right side: 18.927493231 Break this problem into two subproblems by setting (n + -1.5) equal to 18.927493231 and -18.927493231.Subproblem 1
n + -1.5 = 18.927493231 Simplifying n + -1.5 = 18.927493231 Reorder the terms: -1.5 + n = 18.927493231 Solving -1.5 + n = 18.927493231 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 = 18.927493231 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 18.927493231 + 1.5 n = 18.927493231 + 1.5 Combine like terms: 18.927493231 + 1.5 = 20.427493231 n = 20.427493231 Simplifying n = 20.427493231Subproblem 2
n + -1.5 = -18.927493231 Simplifying n + -1.5 = -18.927493231 Reorder the terms: -1.5 + n = -18.927493231 Solving -1.5 + n = -18.927493231 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 = -18.927493231 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -18.927493231 + 1.5 n = -18.927493231 + 1.5 Combine like terms: -18.927493231 + 1.5 = -17.427493231 n = -17.427493231 Simplifying n = -17.427493231Solution
The solution to the problem is based on the solutions from the subproblems. n = {20.427493231, -17.427493231}Solution
n = {0, 20.427493231, -17.427493231}
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