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Simplifying n4 + -4n3 + 3n2 + -5n4 + n3 = 0 Reorder the terms: 3n2 + -4n3 + n3 + n4 + -5n4 = 0 Combine like terms: -4n3 + n3 = -3n3 3n2 + -3n3 + n4 + -5n4 = 0 Combine like terms: n4 + -5n4 = -4n4 3n2 + -3n3 + -4n4 = 0 Solving 3n2 + -3n3 + -4n4 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n2'. n2(3 + -3n + -4n2) = 0Subproblem 1
Set the factor 'n2' equal to zero and attempt to solve: Simplifying n2 = 0 Solving n2 = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n2 = 0 Take the square root of each side: n = {0}Subproblem 2
Set the factor '(3 + -3n + -4n2)' equal to zero and attempt to solve: Simplifying 3 + -3n + -4n2 = 0 Solving 3 + -3n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 0.75n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.75n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.75n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.75n + n2 = 0 + 0.75 0.75n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.75n + n2 = 0.75 The n term is 0.75n. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75n + 0.140625 + n2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + 0.75n + n2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + 0.75n + n2 = 0.890625 Factor a perfect square on the left side: (n + 0.375)(n + 0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (n + 0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
n + 0.375 = 0.943729304 Simplifying n + 0.375 = 0.943729304 Reorder the terms: 0.375 + n = 0.943729304 Solving 0.375 + n = 0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + n = 0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = 0.943729304 + -0.375 n = 0.943729304 + -0.375 Combine like terms: 0.943729304 + -0.375 = 0.568729304 n = 0.568729304 Simplifying n = 0.568729304Subproblem 2
n + 0.375 = -0.943729304 Simplifying n + 0.375 = -0.943729304 Reorder the terms: 0.375 + n = -0.943729304 Solving 0.375 + n = -0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + n = -0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + n = -0.943729304 + -0.375 n = -0.943729304 + -0.375 Combine like terms: -0.943729304 + -0.375 = -1.318729304 n = -1.318729304 Simplifying n = -1.318729304Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.568729304, -1.318729304}Solution
n = {0, 0.568729304, -1.318729304}
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