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Simplifying 11n2 + 1 = 8n Reorder the terms: 1 + 11n2 = 8n Solving 1 + 11n2 = 8n Solving for variable 'n'. Reorder the terms: 1 + -8n + 11n2 = 8n + -8n Combine like terms: 8n + -8n = 0 1 + -8n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 0.09090909091 + -0.7272727273n + n2 = 0 Move the constant term to the right: Add '-0.09090909091' to each side of the equation. 0.09090909091 + -0.7272727273n + -0.09090909091 + n2 = 0 + -0.09090909091 Reorder the terms: 0.09090909091 + -0.09090909091 + -0.7272727273n + n2 = 0 + -0.09090909091 Combine like terms: 0.09090909091 + -0.09090909091 = 0.00000000000 0.00000000000 + -0.7272727273n + n2 = 0 + -0.09090909091 -0.7272727273n + n2 = 0 + -0.09090909091 Combine like terms: 0 + -0.09090909091 = -0.09090909091 -0.7272727273n + n2 = -0.09090909091 The n term is -0.7272727273n. Take half its coefficient (-0.3636363637). Square it (0.1322314050) and add it to both sides. Add '0.1322314050' to each side of the equation. -0.7272727273n + 0.1322314050 + n2 = -0.09090909091 + 0.1322314050 Reorder the terms: 0.1322314050 + -0.7272727273n + n2 = -0.09090909091 + 0.1322314050 Combine like terms: -0.09090909091 + 0.1322314050 = 0.04132231409 0.1322314050 + -0.7272727273n + n2 = 0.04132231409 Factor a perfect square on the left side: (n + -0.3636363637)(n + -0.3636363637) = 0.04132231409 Calculate the square root of the right side: 0.203278907 Break this problem into two subproblems by setting (n + -0.3636363637) equal to 0.203278907 and -0.203278907.Subproblem 1
n + -0.3636363637 = 0.203278907 Simplifying n + -0.3636363637 = 0.203278907 Reorder the terms: -0.3636363637 + n = 0.203278907 Solving -0.3636363637 + n = 0.203278907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3636363637' to each side of the equation. -0.3636363637 + 0.3636363637 + n = 0.203278907 + 0.3636363637 Combine like terms: -0.3636363637 + 0.3636363637 = 0.0000000000 0.0000000000 + n = 0.203278907 + 0.3636363637 n = 0.203278907 + 0.3636363637 Combine like terms: 0.203278907 + 0.3636363637 = 0.5669152707 n = 0.5669152707 Simplifying n = 0.5669152707Subproblem 2
n + -0.3636363637 = -0.203278907 Simplifying n + -0.3636363637 = -0.203278907 Reorder the terms: -0.3636363637 + n = -0.203278907 Solving -0.3636363637 + n = -0.203278907 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.3636363637' to each side of the equation. -0.3636363637 + 0.3636363637 + n = -0.203278907 + 0.3636363637 Combine like terms: -0.3636363637 + 0.3636363637 = 0.0000000000 0.0000000000 + n = -0.203278907 + 0.3636363637 n = -0.203278907 + 0.3636363637 Combine like terms: -0.203278907 + 0.3636363637 = 0.1603574567 n = 0.1603574567 Simplifying n = 0.1603574567Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.5669152707, 0.1603574567}
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