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Simplifying n3 + 5n2 + -4n = 0 Reorder the terms: -4n + 5n2 + n3 = 0 Solving -4n + 5n2 + n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(-4 + 5n + n2) = 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 '(-4 + 5n + n2)' equal to zero and attempt to solve: Simplifying -4 + 5n + n2 = 0 Solving -4 + 5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + 5n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + 5n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 5n + n2 = 0 + 4 5n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 5n + n2 = 4 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 4 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 4 + 6.25 Combine like terms: 4 + 6.25 = 10.25 6.25 + 5n + n2 = 10.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (n + 2.5) equal to 3.201562119 and -3.201562119.Subproblem 1
n + 2.5 = 3.201562119 Simplifying n + 2.5 = 3.201562119 Reorder the terms: 2.5 + n = 3.201562119 Solving 2.5 + n = 3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 3.201562119 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 3.201562119 + -2.5 n = 3.201562119 + -2.5 Combine like terms: 3.201562119 + -2.5 = 0.701562119 n = 0.701562119 Simplifying n = 0.701562119Subproblem 2
n + 2.5 = -3.201562119 Simplifying n + 2.5 = -3.201562119 Reorder the terms: 2.5 + n = -3.201562119 Solving 2.5 + n = -3.201562119 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -3.201562119 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -3.201562119 + -2.5 n = -3.201562119 + -2.5 Combine like terms: -3.201562119 + -2.5 = -5.701562119 n = -5.701562119 Simplifying n = -5.701562119Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.701562119, -5.701562119}Solution
n = {0, 0.701562119, -5.701562119}
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