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Simplifying 90 + (n + 18) * 2n = 180 Reorder the terms: 90 + (18 + n) * 2n = 180 Reorder the terms for easier multiplication: 90 + 2n(18 + n) = 180 90 + (18 * 2n + n * 2n) = 180 90 + (36n + 2n2) = 180 Solving 90 + 36n + 2n2 = 180 Solving for variable 'n'. Reorder the terms: 90 + -180 + 36n + 2n2 = 180 + -180 Combine like terms: 90 + -180 = -90 -90 + 36n + 2n2 = 180 + -180 Combine like terms: 180 + -180 = 0 -90 + 36n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + 18n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + 18n + n2)' equal to zero and attempt to solve: Simplifying -45 + 18n + n2 = 0 Solving -45 + 18n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '45' to each side of the equation. -45 + 18n + 45 + n2 = 0 + 45 Reorder the terms: -45 + 45 + 18n + n2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 18n + n2 = 0 + 45 18n + n2 = 0 + 45 Combine like terms: 0 + 45 = 45 18n + n2 = 45 The n term is 18n. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18n + 81 + n2 = 45 + 81 Reorder the terms: 81 + 18n + n2 = 45 + 81 Combine like terms: 45 + 81 = 126 81 + 18n + n2 = 126 Factor a perfect square on the left side: (n + 9)(n + 9) = 126 Calculate the square root of the right side: 11.22497216 Break this problem into two subproblems by setting (n + 9) equal to 11.22497216 and -11.22497216.Subproblem 1
n + 9 = 11.22497216 Simplifying n + 9 = 11.22497216 Reorder the terms: 9 + n = 11.22497216 Solving 9 + n = 11.22497216 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = 11.22497216 + -9 Combine like terms: 9 + -9 = 0 0 + n = 11.22497216 + -9 n = 11.22497216 + -9 Combine like terms: 11.22497216 + -9 = 2.22497216 n = 2.22497216 Simplifying n = 2.22497216Subproblem 2
n + 9 = -11.22497216 Simplifying n + 9 = -11.22497216 Reorder the terms: 9 + n = -11.22497216 Solving 9 + n = -11.22497216 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = -11.22497216 + -9 Combine like terms: 9 + -9 = 0 0 + n = -11.22497216 + -9 n = -11.22497216 + -9 Combine like terms: -11.22497216 + -9 = -20.22497216 n = -20.22497216 Simplifying n = -20.22497216Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.22497216, -20.22497216}Solution
n = {2.22497216, -20.22497216}
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