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Simplifying 38n4 + -16n5 + -9n3 = 0 Reorder the terms: -9n3 + 38n4 + -16n5 = 0 Solving -9n3 + 38n4 + -16n5 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n3'. n3(-9 + 38n + -16n2) = 0Subproblem 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 '(-9 + 38n + -16n2)' equal to zero and attempt to solve: Simplifying -9 + 38n + -16n2 = 0 Solving -9 + 38n + -16n2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.5625 + -2.375n + n2 = 0 Move the constant term to the right: Add '-0.5625' to each side of the equation. 0.5625 + -2.375n + -0.5625 + n2 = 0 + -0.5625 Reorder the terms: 0.5625 + -0.5625 + -2.375n + n2 = 0 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + -2.375n + n2 = 0 + -0.5625 -2.375n + n2 = 0 + -0.5625 Combine like terms: 0 + -0.5625 = -0.5625 -2.375n + n2 = -0.5625 The n term is -2.375n. Take half its coefficient (-1.1875). Square it (1.41015625) and add it to both sides. Add '1.41015625' to each side of the equation. -2.375n + 1.41015625 + n2 = -0.5625 + 1.41015625 Reorder the terms: 1.41015625 + -2.375n + n2 = -0.5625 + 1.41015625 Combine like terms: -0.5625 + 1.41015625 = 0.84765625 1.41015625 + -2.375n + n2 = 0.84765625 Factor a perfect square on the left side: (n + -1.1875)(n + -1.1875) = 0.84765625 Calculate the square root of the right side: 0.920682491 Break this problem into two subproblems by setting (n + -1.1875) equal to 0.920682491 and -0.920682491.Subproblem 1
n + -1.1875 = 0.920682491 Simplifying n + -1.1875 = 0.920682491 Reorder the terms: -1.1875 + n = 0.920682491 Solving -1.1875 + n = 0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = 0.920682491 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = 0.920682491 + 1.1875 n = 0.920682491 + 1.1875 Combine like terms: 0.920682491 + 1.1875 = 2.108182491 n = 2.108182491 Simplifying n = 2.108182491Subproblem 2
n + -1.1875 = -0.920682491 Simplifying n + -1.1875 = -0.920682491 Reorder the terms: -1.1875 + n = -0.920682491 Solving -1.1875 + n = -0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.1875' to each side of the equation. -1.1875 + 1.1875 + n = -0.920682491 + 1.1875 Combine like terms: -1.1875 + 1.1875 = 0.0000 0.0000 + n = -0.920682491 + 1.1875 n = -0.920682491 + 1.1875 Combine like terms: -0.920682491 + 1.1875 = 0.266817509 n = 0.266817509 Simplifying n = 0.266817509Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.108182491, 0.266817509}Solution
n = {2.108182491, 0.266817509}
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