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Simplifying (3n2 + -4n3 + -4n) + -1(n3 + -6n2 + -2n) = 0 Reorder the terms: (-4n + 3n2 + -4n3) + -1(n3 + -6n2 + -2n) = 0 Remove parenthesis around (-4n + 3n2 + -4n3) -4n + 3n2 + -4n3 + -1(n3 + -6n2 + -2n) = 0 Reorder the terms: -4n + 3n2 + -4n3 + -1(-2n + -6n2 + n3) = 0 -4n + 3n2 + -4n3 + (-2n * -1 + -6n2 * -1 + n3 * -1) = 0 -4n + 3n2 + -4n3 + (2n + 6n2 + -1n3) = 0 Reorder the terms: -4n + 2n + 3n2 + 6n2 + -4n3 + -1n3 = 0 Combine like terms: -4n + 2n = -2n -2n + 3n2 + 6n2 + -4n3 + -1n3 = 0 Combine like terms: 3n2 + 6n2 = 9n2 -2n + 9n2 + -4n3 + -1n3 = 0 Combine like terms: -4n3 + -1n3 = -5n3 -2n + 9n2 + -5n3 = 0 Solving -2n + 9n2 + -5n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(-2 + 9n + -5n2) = 0Subproblem 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 '(-2 + 9n + -5n2)' equal to zero and attempt to solve: Simplifying -2 + 9n + -5n2 = 0 Solving -2 + 9n + -5n2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 0.4 + -1.8n + n2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + -1.8n + -0.4 + n2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + -1.8n + n2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + -1.8n + n2 = 0 + -0.4 -1.8n + n2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 -1.8n + n2 = -0.4 The n term is -1.8n. Take half its coefficient (-0.9). Square it (0.81) and add it to both sides. Add '0.81' to each side of the equation. -1.8n + 0.81 + n2 = -0.4 + 0.81 Reorder the terms: 0.81 + -1.8n + n2 = -0.4 + 0.81 Combine like terms: -0.4 + 0.81 = 0.41 0.81 + -1.8n + n2 = 0.41 Factor a perfect square on the left side: (n + -0.9)(n + -0.9) = 0.41 Calculate the square root of the right side: 0.640312424 Break this problem into two subproblems by setting (n + -0.9) equal to 0.640312424 and -0.640312424.Subproblem 1
n + -0.9 = 0.640312424 Simplifying n + -0.9 = 0.640312424 Reorder the terms: -0.9 + n = 0.640312424 Solving -0.9 + n = 0.640312424 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = 0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = 0.640312424 + 0.9 n = 0.640312424 + 0.9 Combine like terms: 0.640312424 + 0.9 = 1.540312424 n = 1.540312424 Simplifying n = 1.540312424Subproblem 2
n + -0.9 = -0.640312424 Simplifying n + -0.9 = -0.640312424 Reorder the terms: -0.9 + n = -0.640312424 Solving -0.9 + n = -0.640312424 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + n = -0.640312424 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + n = -0.640312424 + 0.9 n = -0.640312424 + 0.9 Combine like terms: -0.640312424 + 0.9 = 0.259687576 n = 0.259687576 Simplifying n = 0.259687576Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.540312424, 0.259687576}Solution
n = {0, 1.540312424, 0.259687576}
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