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Simplifying 30n3 + -48n2 + -96n = 0 Reorder the terms: -96n + -48n2 + 30n3 = 0 Solving -96n + -48n2 + 30n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '6n'. 6n(-16 + -8n + 5n2) = 0 Ignore the factor 6.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 '(-16 + -8n + 5n2)' equal to zero and attempt to solve: Simplifying -16 + -8n + 5n2 = 0 Solving -16 + -8n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -3.2 + -1.6n + n2 = 0 Move the constant term to the right: Add '3.2' to each side of the equation. -3.2 + -1.6n + 3.2 + n2 = 0 + 3.2 Reorder the terms: -3.2 + 3.2 + -1.6n + n2 = 0 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + -1.6n + n2 = 0 + 3.2 -1.6n + n2 = 0 + 3.2 Combine like terms: 0 + 3.2 = 3.2 -1.6n + n2 = 3.2 The n term is -1.6n. Take half its coefficient (-0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. -1.6n + 0.64 + n2 = 3.2 + 0.64 Reorder the terms: 0.64 + -1.6n + n2 = 3.2 + 0.64 Combine like terms: 3.2 + 0.64 = 3.84 0.64 + -1.6n + n2 = 3.84 Factor a perfect square on the left side: (n + -0.8)(n + -0.8) = 3.84 Calculate the square root of the right side: 1.959591794 Break this problem into two subproblems by setting (n + -0.8) equal to 1.959591794 and -1.959591794.Subproblem 1
n + -0.8 = 1.959591794 Simplifying n + -0.8 = 1.959591794 Reorder the terms: -0.8 + n = 1.959591794 Solving -0.8 + n = 1.959591794 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + n = 1.959591794 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + n = 1.959591794 + 0.8 n = 1.959591794 + 0.8 Combine like terms: 1.959591794 + 0.8 = 2.759591794 n = 2.759591794 Simplifying n = 2.759591794Subproblem 2
n + -0.8 = -1.959591794 Simplifying n + -0.8 = -1.959591794 Reorder the terms: -0.8 + n = -1.959591794 Solving -0.8 + n = -1.959591794 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + n = -1.959591794 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + n = -1.959591794 + 0.8 n = -1.959591794 + 0.8 Combine like terms: -1.959591794 + 0.8 = -1.159591794 n = -1.159591794 Simplifying n = -1.159591794Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.759591794, -1.159591794}Solution
n = {0, 2.759591794, -1.159591794}
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