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Simplifying 11n2 = 3 + -2n Solving 11n2 = 3 + -2n Solving for variable 'n'. Reorder the terms: -3 + 2n + 11n2 = 3 + -2n + -3 + 2n Reorder the terms: -3 + 2n + 11n2 = 3 + -3 + -2n + 2n Combine like terms: 3 + -3 = 0 -3 + 2n + 11n2 = 0 + -2n + 2n -3 + 2n + 11n2 = -2n + 2n Combine like terms: -2n + 2n = 0 -3 + 2n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.2727272727 + 0.1818181818n + n2 = 0 Move the constant term to the right: Add '0.2727272727' to each side of the equation. -0.2727272727 + 0.1818181818n + 0.2727272727 + n2 = 0 + 0.2727272727 Reorder the terms: -0.2727272727 + 0.2727272727 + 0.1818181818n + n2 = 0 + 0.2727272727 Combine like terms: -0.2727272727 + 0.2727272727 = 0.0000000000 0.0000000000 + 0.1818181818n + n2 = 0 + 0.2727272727 0.1818181818n + n2 = 0 + 0.2727272727 Combine like terms: 0 + 0.2727272727 = 0.2727272727 0.1818181818n + n2 = 0.2727272727 The n term is 0.1818181818n. Take half its coefficient (0.0909090909). Square it (0.008264462808) and add it to both sides. Add '0.008264462808' to each side of the equation. 0.1818181818n + 0.008264462808 + n2 = 0.2727272727 + 0.008264462808 Reorder the terms: 0.008264462808 + 0.1818181818n + n2 = 0.2727272727 + 0.008264462808 Combine like terms: 0.2727272727 + 0.008264462808 = 0.280991735508 0.008264462808 + 0.1818181818n + n2 = 0.280991735508 Factor a perfect square on the left side: (n + 0.0909090909)(n + 0.0909090909) = 0.280991735508 Calculate the square root of the right side: 0.530086536 Break this problem into two subproblems by setting (n + 0.0909090909) equal to 0.530086536 and -0.530086536.Subproblem 1
n + 0.0909090909 = 0.530086536 Simplifying n + 0.0909090909 = 0.530086536 Reorder the terms: 0.0909090909 + n = 0.530086536 Solving 0.0909090909 + n = 0.530086536 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + n = 0.530086536 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + n = 0.530086536 + -0.0909090909 n = 0.530086536 + -0.0909090909 Combine like terms: 0.530086536 + -0.0909090909 = 0.4391774451 n = 0.4391774451 Simplifying n = 0.4391774451Subproblem 2
n + 0.0909090909 = -0.530086536 Simplifying n + 0.0909090909 = -0.530086536 Reorder the terms: 0.0909090909 + n = -0.530086536 Solving 0.0909090909 + n = -0.530086536 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + n = -0.530086536 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + n = -0.530086536 + -0.0909090909 n = -0.530086536 + -0.0909090909 Combine like terms: -0.530086536 + -0.0909090909 = -0.6209956269 n = -0.6209956269 Simplifying n = -0.6209956269Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.4391774451, -0.6209956269}
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