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Simplifying 8n + -2(2n + 1)(2n + 6) = 11n Reorder the terms: 8n + -2(1 + 2n)(2n + 6) = 11n Reorder the terms: 8n + -2(1 + 2n)(6 + 2n) = 11n Multiply (1 + 2n) * (6 + 2n) 8n + -2(1(6 + 2n) + 2n * (6 + 2n)) = 11n 8n + -2((6 * 1 + 2n * 1) + 2n * (6 + 2n)) = 11n 8n + -2((6 + 2n) + 2n * (6 + 2n)) = 11n 8n + -2(6 + 2n + (6 * 2n + 2n * 2n)) = 11n 8n + -2(6 + 2n + (12n + 4n2)) = 11n Combine like terms: 2n + 12n = 14n 8n + -2(6 + 14n + 4n2) = 11n 8n + (6 * -2 + 14n * -2 + 4n2 * -2) = 11n 8n + (-12 + -28n + -8n2) = 11n Reorder the terms: -12 + 8n + -28n + -8n2 = 11n Combine like terms: 8n + -28n = -20n -12 + -20n + -8n2 = 11n Solving -12 + -20n + -8n2 = 11n Solving for variable 'n'. Reorder the terms: -12 + -20n + -11n + -8n2 = 11n + -11n Combine like terms: -20n + -11n = -31n -12 + -31n + -8n2 = 11n + -11n Combine like terms: 11n + -11n = 0 -12 + -31n + -8n2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(12 + 31n + 8n2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(12 + 31n + 8n2)' equal to zero and attempt to solve: Simplifying 12 + 31n + 8n2 = 0 Solving 12 + 31n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.5 + 3.875n + n2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3.875n + -1.5 + n2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3.875n + n2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3.875n + n2 = 0 + -1.5 3.875n + n2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3.875n + n2 = -1.5 The n term is 3.875n. Take half its coefficient (1.9375). Square it (3.75390625) and add it to both sides. Add '3.75390625' to each side of the equation. 3.875n + 3.75390625 + n2 = -1.5 + 3.75390625 Reorder the terms: 3.75390625 + 3.875n + n2 = -1.5 + 3.75390625 Combine like terms: -1.5 + 3.75390625 = 2.25390625 3.75390625 + 3.875n + n2 = 2.25390625 Factor a perfect square on the left side: (n + 1.9375)(n + 1.9375) = 2.25390625 Calculate the square root of the right side: 1.501301519 Break this problem into two subproblems by setting (n + 1.9375) equal to 1.501301519 and -1.501301519.Subproblem 1
n + 1.9375 = 1.501301519 Simplifying n + 1.9375 = 1.501301519 Reorder the terms: 1.9375 + n = 1.501301519 Solving 1.9375 + n = 1.501301519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + n = 1.501301519 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + n = 1.501301519 + -1.9375 n = 1.501301519 + -1.9375 Combine like terms: 1.501301519 + -1.9375 = -0.436198481 n = -0.436198481 Simplifying n = -0.436198481Subproblem 2
n + 1.9375 = -1.501301519 Simplifying n + 1.9375 = -1.501301519 Reorder the terms: 1.9375 + n = -1.501301519 Solving 1.9375 + n = -1.501301519 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + n = -1.501301519 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + n = -1.501301519 + -1.9375 n = -1.501301519 + -1.9375 Combine like terms: -1.501301519 + -1.9375 = -3.438801519 n = -3.438801519 Simplifying n = -3.438801519Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.436198481, -3.438801519}Solution
n = {-0.436198481, -3.438801519}
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