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Simplifying 12 + (8 + -1n) * 3n = 5 Reorder the terms for easier multiplication: 12 + 3n(8 + -1n) = 5 12 + (8 * 3n + -1n * 3n) = 5 12 + (24n + -3n2) = 5 Solving 12 + 24n + -3n2 = 5 Solving for variable 'n'. Reorder the terms: 12 + -5 + 24n + -3n2 = 5 + -5 Combine like terms: 12 + -5 = 7 7 + 24n + -3n2 = 5 + -5 Combine like terms: 5 + -5 = 0 7 + 24n + -3n2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2.333333333 + -8n + n2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + -8n + 2.333333333 + n2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + -8n + n2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + -8n + n2 = 0 + 2.333333333 -8n + n2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 -8n + n2 = 2.333333333 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 = 2.333333333 + 16 Reorder the terms: 16 + -8n + n2 = 2.333333333 + 16 Combine like terms: 2.333333333 + 16 = 18.333333333 16 + -8n + n2 = 18.333333333 Factor a perfect square on the left side: (n + -4)(n + -4) = 18.333333333 Calculate the square root of the right side: 4.281744193 Break this problem into two subproblems by setting (n + -4) equal to 4.281744193 and -4.281744193.Subproblem 1
n + -4 = 4.281744193 Simplifying n + -4 = 4.281744193 Reorder the terms: -4 + n = 4.281744193 Solving -4 + n = 4.281744193 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 = 4.281744193 + 4 Combine like terms: -4 + 4 = 0 0 + n = 4.281744193 + 4 n = 4.281744193 + 4 Combine like terms: 4.281744193 + 4 = 8.281744193 n = 8.281744193 Simplifying n = 8.281744193Subproblem 2
n + -4 = -4.281744193 Simplifying n + -4 = -4.281744193 Reorder the terms: -4 + n = -4.281744193 Solving -4 + n = -4.281744193 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 = -4.281744193 + 4 Combine like terms: -4 + 4 = 0 0 + n = -4.281744193 + 4 n = -4.281744193 + 4 Combine like terms: -4.281744193 + 4 = -0.281744193 n = -0.281744193 Simplifying n = -0.281744193Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.281744193, -0.281744193}
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