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Simplifying 8n(2n2 + 6n + -10) = 0 Reorder the terms: 8n(-10 + 6n + 2n2) = 0 (-10 * 8n + 6n * 8n + 2n2 * 8n) = 0 (-80n + 48n2 + 16n3) = 0 Solving -80n + 48n2 + 16n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '16n'. 16n(-5 + 3n + n2) = 0 Ignore the factor 16.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 '(-5 + 3n + n2)' equal to zero and attempt to solve: Simplifying -5 + 3n + n2 = 0 Solving -5 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '5' to each side of the equation. -5 + 3n + 5 + n2 = 0 + 5 Reorder the terms: -5 + 5 + 3n + n2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + 3n + n2 = 0 + 5 3n + n2 = 0 + 5 Combine like terms: 0 + 5 = 5 3n + n2 = 5 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 = 5 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + 3n + n2 = 7.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (n + 1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
n + 1.5 = 2.692582404 Simplifying n + 1.5 = 2.692582404 Reorder the terms: 1.5 + n = 2.692582404 Solving 1.5 + n = 2.692582404 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 = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 2.692582404 + -1.5 n = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 n = 1.192582404 Simplifying n = 1.192582404Subproblem 2
n + 1.5 = -2.692582404 Simplifying n + 1.5 = -2.692582404 Reorder the terms: 1.5 + n = -2.692582404 Solving 1.5 + n = -2.692582404 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 = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -2.692582404 + -1.5 n = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 n = -4.192582404 Simplifying n = -4.192582404Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.192582404, -4.192582404}Solution
n = {0, 1.192582404, -4.192582404}
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