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Simplifying 11(8 + -1n) * n = 4 Reorder the terms for easier multiplication: 11n(8 + -1n) = 4 (8 * 11n + -1n * 11n) = 4 (88n + -11n2) = 4 Solving 88n + -11n2 = 4 Solving for variable 'n'. Reorder the terms: -4 + 88n + -11n2 = 4 + -4 Combine like terms: 4 + -4 = 0 -4 + 88n + -11n2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. 0.3636363636 + -8n + n2 = 0 Move the constant term to the right: Add '-0.3636363636' to each side of the equation. 0.3636363636 + -8n + -0.3636363636 + n2 = 0 + -0.3636363636 Reorder the terms: 0.3636363636 + -0.3636363636 + -8n + n2 = 0 + -0.3636363636 Combine like terms: 0.3636363636 + -0.3636363636 = 0.0000000000 0.0000000000 + -8n + n2 = 0 + -0.3636363636 -8n + n2 = 0 + -0.3636363636 Combine like terms: 0 + -0.3636363636 = -0.3636363636 -8n + n2 = -0.3636363636 The n term is -8n. Take half its coefficient (-4). Square it (16) and add it to both sides. Add '16' to each side of the equation. -8n + 16 + n2 = -0.3636363636 + 16 Reorder the terms: 16 + -8n + n2 = -0.3636363636 + 16 Combine like terms: -0.3636363636 + 16 = 15.6363636364 16 + -8n + n2 = 15.6363636364 Factor a perfect square on the left side: (n + -4)(n + -4) = 15.6363636364 Calculate the square root of the right side: 3.954284213 Break this problem into two subproblems by setting (n + -4) equal to 3.954284213 and -3.954284213.Subproblem 1
n + -4 = 3.954284213 Simplifying n + -4 = 3.954284213 Reorder the terms: -4 + n = 3.954284213 Solving -4 + n = 3.954284213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 3.954284213 + 4 Combine like terms: -4 + 4 = 0 0 + n = 3.954284213 + 4 n = 3.954284213 + 4 Combine like terms: 3.954284213 + 4 = 7.954284213 n = 7.954284213 Simplifying n = 7.954284213Subproblem 2
n + -4 = -3.954284213 Simplifying n + -4 = -3.954284213 Reorder the terms: -4 + n = -3.954284213 Solving -4 + n = -3.954284213 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -3.954284213 + 4 Combine like terms: -4 + 4 = 0 0 + n = -3.954284213 + 4 n = -3.954284213 + 4 Combine like terms: -3.954284213 + 4 = 0.045715787 n = 0.045715787 Simplifying n = 0.045715787Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.954284213, 0.045715787}
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