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Simplifying 18n2 + 11n + -54 = 12n2 Reorder the terms: -54 + 11n + 18n2 = 12n2 Solving -54 + 11n + 18n2 = 12n2 Solving for variable 'n'. Combine like terms: 18n2 + -12n2 = 6n2 -54 + 11n + 6n2 = 12n2 + -12n2 Combine like terms: 12n2 + -12n2 = 0 -54 + 11n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -9 + 1.833333333n + n2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 1.833333333n + 9 + n2 = 0 + 9 Reorder the terms: -9 + 9 + 1.833333333n + n2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 1.833333333n + n2 = 0 + 9 1.833333333n + n2 = 0 + 9 Combine like terms: 0 + 9 = 9 1.833333333n + n2 = 9 The n term is 1.833333333n. Take half its coefficient (0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. 1.833333333n + 0.8402777775 + n2 = 9 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333n + n2 = 9 + 0.8402777775 Combine like terms: 9 + 0.8402777775 = 9.8402777775 0.8402777775 + 1.833333333n + n2 = 9.8402777775 Factor a perfect square on the left side: (n + 0.9166666665)(n + 0.9166666665) = 9.8402777775 Calculate the square root of the right side: 3.136921704 Break this problem into two subproblems by setting (n + 0.9166666665) equal to 3.136921704 and -3.136921704.Subproblem 1
n + 0.9166666665 = 3.136921704 Simplifying n + 0.9166666665 = 3.136921704 Reorder the terms: 0.9166666665 + n = 3.136921704 Solving 0.9166666665 + n = 3.136921704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + n = 3.136921704 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + n = 3.136921704 + -0.9166666665 n = 3.136921704 + -0.9166666665 Combine like terms: 3.136921704 + -0.9166666665 = 2.2202550375 n = 2.2202550375 Simplifying n = 2.2202550375Subproblem 2
n + 0.9166666665 = -3.136921704 Simplifying n + 0.9166666665 = -3.136921704 Reorder the terms: 0.9166666665 + n = -3.136921704 Solving 0.9166666665 + n = -3.136921704 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + n = -3.136921704 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + n = -3.136921704 + -0.9166666665 n = -3.136921704 + -0.9166666665 Combine like terms: -3.136921704 + -0.9166666665 = -4.0535883705 n = -4.0535883705 Simplifying n = -4.0535883705Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.2202550375, -4.0535883705}
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