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Simplifying 2n2 + 6n + -180 = 0 Reorder the terms: -180 + 6n + 2n2 = 0 Solving -180 + 6n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-90 + 3n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-90 + 3n + n2)' equal to zero and attempt to solve: Simplifying -90 + 3n + n2 = 0 Solving -90 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '90' to each side of the equation. -90 + 3n + 90 + n2 = 0 + 90 Reorder the terms: -90 + 90 + 3n + n2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 3n + n2 = 0 + 90 3n + n2 = 0 + 90 Combine like terms: 0 + 90 = 90 3n + n2 = 90 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 90 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 90 + 2.25 Combine like terms: 90 + 2.25 = 92.25 2.25 + 3n + n2 = 92.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 92.25 Calculate the square root of the right side: 9.604686356 Break this problem into two subproblems by setting (n + 1.5) equal to 9.604686356 and -9.604686356.Subproblem 1
n + 1.5 = 9.604686356 Simplifying n + 1.5 = 9.604686356 Reorder the terms: 1.5 + n = 9.604686356 Solving 1.5 + n = 9.604686356 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 9.604686356 + -1.5 n = 9.604686356 + -1.5 Combine like terms: 9.604686356 + -1.5 = 8.104686356 n = 8.104686356 Simplifying n = 8.104686356Subproblem 2
n + 1.5 = -9.604686356 Simplifying n + 1.5 = -9.604686356 Reorder the terms: 1.5 + n = -9.604686356 Solving 1.5 + n = -9.604686356 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -9.604686356 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -9.604686356 + -1.5 n = -9.604686356 + -1.5 Combine like terms: -9.604686356 + -1.5 = -11.104686356 n = -11.104686356 Simplifying n = -11.104686356Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.104686356, -11.104686356}Solution
n = {8.104686356, -11.104686356}
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