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Simplifying (2n + 6)(4n + 2) + 16 = 80 Reorder the terms: (6 + 2n)(4n + 2) + 16 = 80 Reorder the terms: (6 + 2n)(2 + 4n) + 16 = 80 Multiply (6 + 2n) * (2 + 4n) (6(2 + 4n) + 2n * (2 + 4n)) + 16 = 80 ((2 * 6 + 4n * 6) + 2n * (2 + 4n)) + 16 = 80 ((12 + 24n) + 2n * (2 + 4n)) + 16 = 80 (12 + 24n + (2 * 2n + 4n * 2n)) + 16 = 80 (12 + 24n + (4n + 8n2)) + 16 = 80 Combine like terms: 24n + 4n = 28n (12 + 28n + 8n2) + 16 = 80 Reorder the terms: 12 + 16 + 28n + 8n2 = 80 Combine like terms: 12 + 16 = 28 28 + 28n + 8n2 = 80 Solving 28 + 28n + 8n2 = 80 Solving for variable 'n'. Reorder the terms: 28 + -80 + 28n + 8n2 = 80 + -80 Combine like terms: 28 + -80 = -52 -52 + 28n + 8n2 = 80 + -80 Combine like terms: 80 + -80 = 0 -52 + 28n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-13 + 7n + 2n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-13 + 7n + 2n2)' equal to zero and attempt to solve: Simplifying -13 + 7n + 2n2 = 0 Solving -13 + 7n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + 3.5n + n2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + 3.5n + 6.5 + n2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + 3.5n + n2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + 3.5n + n2 = 0 + 6.5 3.5n + n2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 3.5n + n2 = 6.5 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = 6.5 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = 6.5 + 3.0625 Combine like terms: 6.5 + 3.0625 = 9.5625 3.0625 + 3.5n + n2 = 9.5625 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (n + 1.75) equal to 3.092329219 and -3.092329219.Subproblem 1
n + 1.75 = 3.092329219 Simplifying n + 1.75 = 3.092329219 Reorder the terms: 1.75 + n = 3.092329219 Solving 1.75 + n = 3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 3.092329219 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 3.092329219 + -1.75 n = 3.092329219 + -1.75 Combine like terms: 3.092329219 + -1.75 = 1.342329219 n = 1.342329219 Simplifying n = 1.342329219Subproblem 2
n + 1.75 = -3.092329219 Simplifying n + 1.75 = -3.092329219 Reorder the terms: 1.75 + n = -3.092329219 Solving 1.75 + n = -3.092329219 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -3.092329219 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -3.092329219 + -1.75 n = -3.092329219 + -1.75 Combine like terms: -3.092329219 + -1.75 = -4.842329219 n = -4.842329219 Simplifying n = -4.842329219Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.342329219, -4.842329219}Solution
n = {1.342329219, -4.842329219}
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