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Simplifying 10n3 + -64n2 + 81n = 0 Reorder the terms: 81n + -64n2 + 10n3 = 0 Solving 81n + -64n2 + 10n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(81 + -64n + 10n2) = 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 '(81 + -64n + 10n2)' equal to zero and attempt to solve: Simplifying 81 + -64n + 10n2 = 0 Solving 81 + -64n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 8.1 + -6.4n + n2 = 0 Move the constant term to the right: Add '-8.1' to each side of the equation. 8.1 + -6.4n + -8.1 + n2 = 0 + -8.1 Reorder the terms: 8.1 + -8.1 + -6.4n + n2 = 0 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + -6.4n + n2 = 0 + -8.1 -6.4n + n2 = 0 + -8.1 Combine like terms: 0 + -8.1 = -8.1 -6.4n + n2 = -8.1 The n term is -6.4n. Take half its coefficient (-3.2). Square it (10.24) and add it to both sides. Add '10.24' to each side of the equation. -6.4n + 10.24 + n2 = -8.1 + 10.24 Reorder the terms: 10.24 + -6.4n + n2 = -8.1 + 10.24 Combine like terms: -8.1 + 10.24 = 2.14 10.24 + -6.4n + n2 = 2.14 Factor a perfect square on the left side: (n + -3.2)(n + -3.2) = 2.14 Calculate the square root of the right side: 1.462873884 Break this problem into two subproblems by setting (n + -3.2) equal to 1.462873884 and -1.462873884.Subproblem 1
n + -3.2 = 1.462873884 Simplifying n + -3.2 = 1.462873884 Reorder the terms: -3.2 + n = 1.462873884 Solving -3.2 + n = 1.462873884 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = 1.462873884 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = 1.462873884 + 3.2 n = 1.462873884 + 3.2 Combine like terms: 1.462873884 + 3.2 = 4.662873884 n = 4.662873884 Simplifying n = 4.662873884Subproblem 2
n + -3.2 = -1.462873884 Simplifying n + -3.2 = -1.462873884 Reorder the terms: -3.2 + n = -1.462873884 Solving -3.2 + n = -1.462873884 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = -1.462873884 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = -1.462873884 + 3.2 n = -1.462873884 + 3.2 Combine like terms: -1.462873884 + 3.2 = 1.737126116 n = 1.737126116 Simplifying n = 1.737126116Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.662873884, 1.737126116}Solution
n = {0, 4.662873884, 1.737126116}
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