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Simplifying 180n2 + -360n + -2160 = 0 Reorder the terms: -2160 + -360n + 180n2 = 0 Solving -2160 + -360n + 180n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '180'. 180(-12 + -2n + n2) = 0 Ignore the factor 180.Subproblem 1
Set the factor '(-12 + -2n + n2)' equal to zero and attempt to solve: Simplifying -12 + -2n + n2 = 0 Solving -12 + -2n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + -2n + 12 + n2 = 0 + 12 Reorder the terms: -12 + 12 + -2n + n2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + -2n + n2 = 0 + 12 -2n + n2 = 0 + 12 Combine like terms: 0 + 12 = 12 -2n + n2 = 12 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 12 + 1 Reorder the terms: 1 + -2n + n2 = 12 + 1 Combine like terms: 12 + 1 = 13 1 + -2n + n2 = 13 Factor a perfect square on the left side: (n + -1)(n + -1) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (n + -1) equal to 3.605551275 and -3.605551275.Subproblem 1
n + -1 = 3.605551275 Simplifying n + -1 = 3.605551275 Reorder the terms: -1 + n = 3.605551275 Solving -1 + n = 3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 3.605551275 + 1 Combine like terms: -1 + 1 = 0 0 + n = 3.605551275 + 1 n = 3.605551275 + 1 Combine like terms: 3.605551275 + 1 = 4.605551275 n = 4.605551275 Simplifying n = 4.605551275Subproblem 2
n + -1 = -3.605551275 Simplifying n + -1 = -3.605551275 Reorder the terms: -1 + n = -3.605551275 Solving -1 + n = -3.605551275 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -3.605551275 + 1 Combine like terms: -1 + 1 = 0 0 + n = -3.605551275 + 1 n = -3.605551275 + 1 Combine like terms: -3.605551275 + 1 = -2.605551275 n = -2.605551275 Simplifying n = -2.605551275Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.605551275, -2.605551275}Solution
n = {4.605551275, -2.605551275}
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