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Simplifying 11n2 = 16 + -12n Solving 11n2 = 16 + -12n Solving for variable 'n'. Reorder the terms: -16 + 12n + 11n2 = 16 + -12n + -16 + 12n Reorder the terms: -16 + 12n + 11n2 = 16 + -16 + -12n + 12n Combine like terms: 16 + -16 = 0 -16 + 12n + 11n2 = 0 + -12n + 12n -16 + 12n + 11n2 = -12n + 12n Combine like terms: -12n + 12n = 0 -16 + 12n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.454545455 + 1.090909091n + n2 = 0 Move the constant term to the right: Add '1.454545455' to each side of the equation. -1.454545455 + 1.090909091n + 1.454545455 + n2 = 0 + 1.454545455 Reorder the terms: -1.454545455 + 1.454545455 + 1.090909091n + n2 = 0 + 1.454545455 Combine like terms: -1.454545455 + 1.454545455 = 0.000000000 0.000000000 + 1.090909091n + n2 = 0 + 1.454545455 1.090909091n + n2 = 0 + 1.454545455 Combine like terms: 0 + 1.454545455 = 1.454545455 1.090909091n + n2 = 1.454545455 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 = 1.454545455 + 0.2975206612 Reorder the terms: 0.2975206612 + 1.090909091n + n2 = 1.454545455 + 0.2975206612 Combine like terms: 1.454545455 + 0.2975206612 = 1.7520661162 0.2975206612 + 1.090909091n + n2 = 1.7520661162 Factor a perfect square on the left side: (n + 0.5454545455)(n + 0.5454545455) = 1.7520661162 Calculate the square root of the right side: 1.323656344 Break this problem into two subproblems by setting (n + 0.5454545455) equal to 1.323656344 and -1.323656344.Subproblem 1
n + 0.5454545455 = 1.323656344 Simplifying n + 0.5454545455 = 1.323656344 Reorder the terms: 0.5454545455 + n = 1.323656344 Solving 0.5454545455 + n = 1.323656344 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 = 1.323656344 + -0.5454545455 Combine like terms: 0.5454545455 + -0.5454545455 = 0.0000000000 0.0000000000 + n = 1.323656344 + -0.5454545455 n = 1.323656344 + -0.5454545455 Combine like terms: 1.323656344 + -0.5454545455 = 0.7782017985 n = 0.7782017985 Simplifying n = 0.7782017985Subproblem 2
n + 0.5454545455 = -1.323656344 Simplifying n + 0.5454545455 = -1.323656344 Reorder the terms: 0.5454545455 + n = -1.323656344 Solving 0.5454545455 + n = -1.323656344 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 = -1.323656344 + -0.5454545455 Combine like terms: 0.5454545455 + -0.5454545455 = 0.0000000000 0.0000000000 + n = -1.323656344 + -0.5454545455 n = -1.323656344 + -0.5454545455 Combine like terms: -1.323656344 + -0.5454545455 = -1.8691108895 n = -1.8691108895 Simplifying n = -1.8691108895Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.7782017985, -1.8691108895}
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