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Simplifying 14 + 3n = 8n(n + -4) Reorder the terms: 14 + 3n = 8n(-4 + n) 14 + 3n = (-4 * 8n + n * 8n) 14 + 3n = (-32n + 8n2) Solving 14 + 3n = -32n + 8n2 Solving for variable 'n'. Combine like terms: 3n + 32n = 35n 14 + 35n + -8n2 = -32n + 8n2 + 32n + -8n2 Reorder the terms: 14 + 35n + -8n2 = -32n + 32n + 8n2 + -8n2 Combine like terms: -32n + 32n = 0 14 + 35n + -8n2 = 0 + 8n2 + -8n2 14 + 35n + -8n2 = 8n2 + -8n2 Combine like terms: 8n2 + -8n2 = 0 14 + 35n + -8n2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.75 + -4.375n + n2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + -4.375n + 1.75 + n2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + -4.375n + n2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + -4.375n + n2 = 0 + 1.75 -4.375n + n2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 -4.375n + n2 = 1.75 The n term is -4.375n. Take half its coefficient (-2.1875). Square it (4.78515625) and add it to both sides. Add '4.78515625' to each side of the equation. -4.375n + 4.78515625 + n2 = 1.75 + 4.78515625 Reorder the terms: 4.78515625 + -4.375n + n2 = 1.75 + 4.78515625 Combine like terms: 1.75 + 4.78515625 = 6.53515625 4.78515625 + -4.375n + n2 = 6.53515625 Factor a perfect square on the left side: (n + -2.1875)(n + -2.1875) = 6.53515625 Calculate the square root of the right side: 2.556395167 Break this problem into two subproblems by setting (n + -2.1875) equal to 2.556395167 and -2.556395167.Subproblem 1
n + -2.1875 = 2.556395167 Simplifying n + -2.1875 = 2.556395167 Reorder the terms: -2.1875 + n = 2.556395167 Solving -2.1875 + n = 2.556395167 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + n = 2.556395167 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + n = 2.556395167 + 2.1875 n = 2.556395167 + 2.1875 Combine like terms: 2.556395167 + 2.1875 = 4.743895167 n = 4.743895167 Simplifying n = 4.743895167Subproblem 2
n + -2.1875 = -2.556395167 Simplifying n + -2.1875 = -2.556395167 Reorder the terms: -2.1875 + n = -2.556395167 Solving -2.1875 + n = -2.556395167 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.1875' to each side of the equation. -2.1875 + 2.1875 + n = -2.556395167 + 2.1875 Combine like terms: -2.1875 + 2.1875 = 0.0000 0.0000 + n = -2.556395167 + 2.1875 n = -2.556395167 + 2.1875 Combine like terms: -2.556395167 + 2.1875 = -0.368895167 n = -0.368895167 Simplifying n = -0.368895167Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.743895167, -0.368895167}
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