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Simplifying 2n2 + 38n + -540 = 0 Reorder the terms: -540 + 38n + 2n2 = 0 Solving -540 + 38n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-270 + 19n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-270 + 19n + n2)' equal to zero and attempt to solve: Simplifying -270 + 19n + n2 = 0 Solving -270 + 19n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '270' to each side of the equation. -270 + 19n + 270 + n2 = 0 + 270 Reorder the terms: -270 + 270 + 19n + n2 = 0 + 270 Combine like terms: -270 + 270 = 0 0 + 19n + n2 = 0 + 270 19n + n2 = 0 + 270 Combine like terms: 0 + 270 = 270 19n + n2 = 270 The n term is 19n. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19n + 90.25 + n2 = 270 + 90.25 Reorder the terms: 90.25 + 19n + n2 = 270 + 90.25 Combine like terms: 270 + 90.25 = 360.25 90.25 + 19n + n2 = 360.25 Factor a perfect square on the left side: (n + 9.5)(n + 9.5) = 360.25 Calculate the square root of the right side: 18.980252896 Break this problem into two subproblems by setting (n + 9.5) equal to 18.980252896 and -18.980252896.Subproblem 1
n + 9.5 = 18.980252896 Simplifying n + 9.5 = 18.980252896 Reorder the terms: 9.5 + n = 18.980252896 Solving 9.5 + n = 18.980252896 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = 18.980252896 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = 18.980252896 + -9.5 n = 18.980252896 + -9.5 Combine like terms: 18.980252896 + -9.5 = 9.480252896 n = 9.480252896 Simplifying n = 9.480252896Subproblem 2
n + 9.5 = -18.980252896 Simplifying n + 9.5 = -18.980252896 Reorder the terms: 9.5 + n = -18.980252896 Solving 9.5 + n = -18.980252896 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + n = -18.980252896 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + n = -18.980252896 + -9.5 n = -18.980252896 + -9.5 Combine like terms: -18.980252896 + -9.5 = -28.480252896 n = -28.480252896 Simplifying n = -28.480252896Solution
The solution to the problem is based on the solutions from the subproblems. n = {9.480252896, -28.480252896}Solution
n = {9.480252896, -28.480252896}
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