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Simplifying 2n4 + -27n2 = -23n3 Reorder the terms: -27n2 + 2n4 = -23n3 Solving -27n2 + 2n4 = -23n3 Solving for variable 'n'. Reorder the terms: -27n2 + 23n3 + 2n4 = -23n3 + 23n3 Combine like terms: -23n3 + 23n3 = 0 -27n2 + 23n3 + 2n4 = 0 Factor out the Greatest Common Factor (GCF), 'n2'. n2(-27 + 23n + 2n2) = 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 '(-27 + 23n + 2n2)' equal to zero and attempt to solve: Simplifying -27 + 23n + 2n2 = 0 Solving -27 + 23n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -13.5 + 11.5n + n2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 11.5n + 13.5 + n2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 11.5n + n2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 11.5n + n2 = 0 + 13.5 11.5n + n2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 11.5n + n2 = 13.5 The n term is 11.5n. Take half its coefficient (5.75). Square it (33.0625) and add it to both sides. Add '33.0625' to each side of the equation. 11.5n + 33.0625 + n2 = 13.5 + 33.0625 Reorder the terms: 33.0625 + 11.5n + n2 = 13.5 + 33.0625 Combine like terms: 13.5 + 33.0625 = 46.5625 33.0625 + 11.5n + n2 = 46.5625 Factor a perfect square on the left side: (n + 5.75)(n + 5.75) = 46.5625 Calculate the square root of the right side: 6.823672032 Break this problem into two subproblems by setting (n + 5.75) equal to 6.823672032 and -6.823672032.Subproblem 1
n + 5.75 = 6.823672032 Simplifying n + 5.75 = 6.823672032 Reorder the terms: 5.75 + n = 6.823672032 Solving 5.75 + n = 6.823672032 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = 6.823672032 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = 6.823672032 + -5.75 n = 6.823672032 + -5.75 Combine like terms: 6.823672032 + -5.75 = 1.073672032 n = 1.073672032 Simplifying n = 1.073672032Subproblem 2
n + 5.75 = -6.823672032 Simplifying n + 5.75 = -6.823672032 Reorder the terms: 5.75 + n = -6.823672032 Solving 5.75 + n = -6.823672032 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + n = -6.823672032 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + n = -6.823672032 + -5.75 n = -6.823672032 + -5.75 Combine like terms: -6.823672032 + -5.75 = -12.573672032 n = -12.573672032 Simplifying n = -12.573672032Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.073672032, -12.573672032}Solution
n = {0, 1.073672032, -12.573672032}
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