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Simplifying 2(8n + 2) + -1n(-8 + -3n) = 0 Reorder the terms: 2(2 + 8n) + -1n(-8 + -3n) = 0 (2 * 2 + 8n * 2) + -1n(-8 + -3n) = 0 (4 + 16n) + -1n(-8 + -3n) = 0 4 + 16n + (-8 * -1n + -3n * -1n) = 0 4 + 16n + (8n + 3n2) = 0 Combine like terms: 16n + 8n = 24n 4 + 24n + 3n2 = 0 Solving 4 + 24n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 8n + n2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 8n + -1.333333333 + n2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 8n + n2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 8n + n2 = 0 + -1.333333333 8n + n2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 8n + n2 = -1.333333333 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 = -1.333333333 + 16 Reorder the terms: 16 + 8n + n2 = -1.333333333 + 16 Combine like terms: -1.333333333 + 16 = 14.666666667 16 + 8n + n2 = 14.666666667 Factor a perfect square on the left side: (n + 4)(n + 4) = 14.666666667 Calculate the square root of the right side: 3.829708431 Break this problem into two subproblems by setting (n + 4) equal to 3.829708431 and -3.829708431.Subproblem 1
n + 4 = 3.829708431 Simplifying n + 4 = 3.829708431 Reorder the terms: 4 + n = 3.829708431 Solving 4 + n = 3.829708431 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 = 3.829708431 + -4 Combine like terms: 4 + -4 = 0 0 + n = 3.829708431 + -4 n = 3.829708431 + -4 Combine like terms: 3.829708431 + -4 = -0.170291569 n = -0.170291569 Simplifying n = -0.170291569Subproblem 2
n + 4 = -3.829708431 Simplifying n + 4 = -3.829708431 Reorder the terms: 4 + n = -3.829708431 Solving 4 + n = -3.829708431 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 = -3.829708431 + -4 Combine like terms: 4 + -4 = 0 0 + n = -3.829708431 + -4 n = -3.829708431 + -4 Combine like terms: -3.829708431 + -4 = -7.829708431 n = -7.829708431 Simplifying n = -7.829708431Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.170291569, -7.829708431}
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