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Simplifying n2 = -54 + -18n Solving n2 = -54 + -18n Solving for variable 'n'. Reorder the terms: 54 + 18n + n2 = -54 + -18n + 54 + 18n Reorder the terms: 54 + 18n + n2 = -54 + 54 + -18n + 18n Combine like terms: -54 + 54 = 0 54 + 18n + n2 = 0 + -18n + 18n 54 + 18n + n2 = -18n + 18n Combine like terms: -18n + 18n = 0 54 + 18n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-54' to each side of the equation. 54 + 18n + -54 + n2 = 0 + -54 Reorder the terms: 54 + -54 + 18n + n2 = 0 + -54 Combine like terms: 54 + -54 = 0 0 + 18n + n2 = 0 + -54 18n + n2 = 0 + -54 Combine like terms: 0 + -54 = -54 18n + n2 = -54 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 = -54 + 81 Reorder the terms: 81 + 18n + n2 = -54 + 81 Combine like terms: -54 + 81 = 27 81 + 18n + n2 = 27 Factor a perfect square on the left side: (n + 9)(n + 9) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (n + 9) equal to 5.196152423 and -5.196152423.Subproblem 1
n + 9 = 5.196152423 Simplifying n + 9 = 5.196152423 Reorder the terms: 9 + n = 5.196152423 Solving 9 + n = 5.196152423 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 = 5.196152423 + -9 Combine like terms: 9 + -9 = 0 0 + n = 5.196152423 + -9 n = 5.196152423 + -9 Combine like terms: 5.196152423 + -9 = -3.803847577 n = -3.803847577 Simplifying n = -3.803847577Subproblem 2
n + 9 = -5.196152423 Simplifying n + 9 = -5.196152423 Reorder the terms: 9 + n = -5.196152423 Solving 9 + n = -5.196152423 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 = -5.196152423 + -9 Combine like terms: 9 + -9 = 0 0 + n = -5.196152423 + -9 n = -5.196152423 + -9 Combine like terms: -5.196152423 + -9 = -14.196152423 n = -14.196152423 Simplifying n = -14.196152423Solution
The solution to the problem is based on the solutions from the subproblems. n = {-3.803847577, -14.196152423}
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