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Simplifying 11n2 = 6 + -3n Solving 11n2 = 6 + -3n Solving for variable 'n'. Reorder the terms: -6 + 3n + 11n2 = 6 + -3n + -6 + 3n Reorder the terms: -6 + 3n + 11n2 = 6 + -6 + -3n + 3n Combine like terms: 6 + -6 = 0 -6 + 3n + 11n2 = 0 + -3n + 3n -6 + 3n + 11n2 = -3n + 3n Combine like terms: -3n + 3n = 0 -6 + 3n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.5454545455 + 0.2727272727n + n2 = 0 Move the constant term to the right: Add '0.5454545455' to each side of the equation. -0.5454545455 + 0.2727272727n + 0.5454545455 + n2 = 0 + 0.5454545455 Reorder the terms: -0.5454545455 + 0.5454545455 + 0.2727272727n + n2 = 0 + 0.5454545455 Combine like terms: -0.5454545455 + 0.5454545455 = 0.0000000000 0.0000000000 + 0.2727272727n + n2 = 0 + 0.5454545455 0.2727272727n + n2 = 0 + 0.5454545455 Combine like terms: 0 + 0.5454545455 = 0.5454545455 0.2727272727n + n2 = 0.5454545455 The n term is 0.2727272727n. Take half its coefficient (0.1363636364). Square it (0.01859504133) and add it to both sides. Add '0.01859504133' to each side of the equation. 0.2727272727n + 0.01859504133 + n2 = 0.5454545455 + 0.01859504133 Reorder the terms: 0.01859504133 + 0.2727272727n + n2 = 0.5454545455 + 0.01859504133 Combine like terms: 0.5454545455 + 0.01859504133 = 0.56404958683 0.01859504133 + 0.2727272727n + n2 = 0.56404958683 Factor a perfect square on the left side: (n + 0.1363636364)(n + 0.1363636364) = 0.56404958683 Calculate the square root of the right side: 0.751032347 Break this problem into two subproblems by setting (n + 0.1363636364) equal to 0.751032347 and -0.751032347.Subproblem 1
n + 0.1363636364 = 0.751032347 Simplifying n + 0.1363636364 = 0.751032347 Reorder the terms: 0.1363636364 + n = 0.751032347 Solving 0.1363636364 + n = 0.751032347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + n = 0.751032347 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + n = 0.751032347 + -0.1363636364 n = 0.751032347 + -0.1363636364 Combine like terms: 0.751032347 + -0.1363636364 = 0.6146687106 n = 0.6146687106 Simplifying n = 0.6146687106Subproblem 2
n + 0.1363636364 = -0.751032347 Simplifying n + 0.1363636364 = -0.751032347 Reorder the terms: 0.1363636364 + n = -0.751032347 Solving 0.1363636364 + n = -0.751032347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + n = -0.751032347 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + n = -0.751032347 + -0.1363636364 n = -0.751032347 + -0.1363636364 Combine like terms: -0.751032347 + -0.1363636364 = -0.8873959834 n = -0.8873959834 Simplifying n = -0.8873959834Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.6146687106, -0.8873959834}
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