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Simplifying 12n7 + 3n5 + -30n3 = 0 Reorder the terms: -30n3 + 3n5 + 12n7 = 0 Solving -30n3 + 3n5 + 12n7 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3n3'. 3n3(-10 + n2 + 4n4) = 0 Ignore the factor 3.Subproblem 1
Set the factor 'n3' equal to zero and attempt to solve: Simplifying n3 = 0 Solving n3 = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n3 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(-10 + n2 + 4n4)' equal to zero and attempt to solve: Simplifying -10 + n2 + 4n4 = 0 Solving -10 + n2 + 4n4 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.5 + 0.25n2 + n4 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.25n2 + 2.5 + n4 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.25n2 + n4 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.25n2 + n4 = 0 + 2.5 0.25n2 + n4 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 0.25n2 + n4 = 2.5 The n term is n2. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25n2 + 0.25 + n4 = 2.5 + 0.25 Reorder the terms: 0.25 + 0.25n2 + n4 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 0.25 + 0.25n2 + n4 = 2.75 Factor a perfect square on the left side: (n2 + 0.5)(n2 + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (n2 + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
n2 + 0.5 = 1.658312395 Simplifying n2 + 0.5 = 1.658312395 Reorder the terms: 0.5 + n2 = 1.658312395 Solving 0.5 + n2 = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n2 = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n2 = 1.658312395 + -0.5 n2 = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 n2 = 1.158312395 Simplifying n2 = 1.158312395 Take the square root of each side: n = {-1.076249225, 1.076249225}Subproblem 2
n2 + 0.5 = -1.658312395 Simplifying n2 + 0.5 = -1.658312395 Reorder the terms: 0.5 + n2 = -1.658312395 Solving 0.5 + n2 = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n2 = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n2 = -1.658312395 + -0.5 n2 = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 n2 = -2.158312395 Simplifying n2 = -2.158312395 Reorder the terms: 2.158312395 + n2 = -2.158312395 + 2.158312395 Combine like terms: -2.158312395 + 2.158312395 = 0.000000000 2.158312395 + 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. The solution to this equation could not be determined.
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