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Simplifying 11n2 = -2 + -12n Solving 11n2 = -2 + -12n Solving for variable 'n'. Reorder the terms: 2 + 12n + 11n2 = -2 + -12n + 2 + 12n Reorder the terms: 2 + 12n + 11n2 = -2 + 2 + -12n + 12n Combine like terms: -2 + 2 = 0 2 + 12n + 11n2 = 0 + -12n + 12n 2 + 12n + 11n2 = -12n + 12n Combine like terms: -12n + 12n = 0 2 + 12n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.1818181818 + 1.090909091n + n2 = 0 Move the constant term to the right: Add '-0.1818181818' to each side of the equation. 0.1818181818 + 1.090909091n + -0.1818181818 + n2 = 0 + -0.1818181818 Reorder the terms: 0.1818181818 + -0.1818181818 + 1.090909091n + n2 = 0 + -0.1818181818 Combine like terms: 0.1818181818 + -0.1818181818 = 0.0000000000 0.0000000000 + 1.090909091n + n2 = 0 + -0.1818181818 1.090909091n + n2 = 0 + -0.1818181818 Combine like terms: 0 + -0.1818181818 = -0.1818181818 1.090909091n + n2 = -0.1818181818 The n term is 1.090909091n. Take half its coefficient (0.5454545455). Square it (0.2975206612) and add it to both sides. Add '0.2975206612' to each side of the equation. 1.090909091n + 0.2975206612 + n2 = -0.1818181818 + 0.2975206612 Reorder the terms: 0.2975206612 + 1.090909091n + n2 = -0.1818181818 + 0.2975206612 Combine like terms: -0.1818181818 + 0.2975206612 = 0.1157024794 0.2975206612 + 1.090909091n + n2 = 0.1157024794 Factor a perfect square on the left side: (n + 0.5454545455)(n + 0.5454545455) = 0.1157024794 Calculate the square root of the right side: 0.340150672 Break this problem into two subproblems by setting (n + 0.5454545455) equal to 0.340150672 and -0.340150672.Subproblem 1
n + 0.5454545455 = 0.340150672 Simplifying n + 0.5454545455 = 0.340150672 Reorder the terms: 0.5454545455 + n = 0.340150672 Solving 0.5454545455 + n = 0.340150672 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5454545455' to each side of the equation. 0.5454545455 + -0.5454545455 + n = 0.340150672 + -0.5454545455 Combine like terms: 0.5454545455 + -0.5454545455 = 0.0000000000 0.0000000000 + n = 0.340150672 + -0.5454545455 n = 0.340150672 + -0.5454545455 Combine like terms: 0.340150672 + -0.5454545455 = -0.2053038735 n = -0.2053038735 Simplifying n = -0.2053038735Subproblem 2
n + 0.5454545455 = -0.340150672 Simplifying n + 0.5454545455 = -0.340150672 Reorder the terms: 0.5454545455 + n = -0.340150672 Solving 0.5454545455 + n = -0.340150672 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5454545455' to each side of the equation. 0.5454545455 + -0.5454545455 + n = -0.340150672 + -0.5454545455 Combine like terms: 0.5454545455 + -0.5454545455 = 0.0000000000 0.0000000000 + n = -0.340150672 + -0.5454545455 n = -0.340150672 + -0.5454545455 Combine like terms: -0.340150672 + -0.5454545455 = -0.8856052175 n = -0.8856052175 Simplifying n = -0.8856052175Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.2053038735, -0.8856052175}
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