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Simplifying -8n5 + 5n3 + 6n = 0 Reorder the terms: 6n + 5n3 + -8n5 = 0 Solving 6n + 5n3 + -8n5 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(6 + 5n2 + -8n4) = 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 '(6 + 5n2 + -8n4)' equal to zero and attempt to solve: Simplifying 6 + 5n2 + -8n4 = 0 Solving 6 + 5n2 + -8n4 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.75 + -0.625n2 + n4 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.625n2 + 0.75 + n4 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.625n2 + n4 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.625n2 + n4 = 0 + 0.75 -0.625n2 + n4 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.625n2 + n4 = 0.75 The n term is -0.625n2. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625n2 + 0.09765625 + n4 = 0.75 + 0.09765625 Reorder the terms: 0.09765625 + -0.625n2 + n4 = 0.75 + 0.09765625 Combine like terms: 0.75 + 0.09765625 = 0.84765625 0.09765625 + -0.625n2 + n4 = 0.84765625 Factor a perfect square on the left side: (n2 + -0.3125)(n2 + -0.3125) = 0.84765625 Calculate the square root of the right side: 0.920682491 Break this problem into two subproblems by setting (n2 + -0.3125) equal to 0.920682491 and -0.920682491.Subproblem 1
n2 + -0.3125 = 0.920682491 Simplifying n2 + -0.3125 = 0.920682491 Reorder the terms: -0.3125 + n2 = 0.920682491 Solving -0.3125 + n2 = 0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + n2 = 0.920682491 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + n2 = 0.920682491 + 0.3125 n2 = 0.920682491 + 0.3125 Combine like terms: 0.920682491 + 0.3125 = 1.233182491 n2 = 1.233182491 Simplifying n2 = 1.233182491 Take the square root of each side: n = {-1.110487502, 1.110487502}Subproblem 2
n2 + -0.3125 = -0.920682491 Simplifying n2 + -0.3125 = -0.920682491 Reorder the terms: -0.3125 + n2 = -0.920682491 Solving -0.3125 + n2 = -0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + n2 = -0.920682491 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + n2 = -0.920682491 + 0.3125 n2 = -0.920682491 + 0.3125 Combine like terms: -0.920682491 + 0.3125 = -0.608182491 n2 = -0.608182491 Simplifying n2 = -0.608182491 Reorder the terms: 0.608182491 + n2 = -0.608182491 + 0.608182491 Combine like terms: -0.608182491 + 0.608182491 = 0.000000000 0.608182491 + n2 = 0.000000000 The solution to this equation could not be determined.This subproblem is being ignored because a solution could not be determined. The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Solution
n = {0}
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