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Simplifying -3(n + -11) = -5(n + 9) * 8n Reorder the terms: -3(-11 + n) = -5(n + 9) * 8n (-11 * -3 + n * -3) = -5(n + 9) * 8n (33 + -3n) = -5(n + 9) * 8n Reorder the terms: 33 + -3n = -5(9 + n) * 8n Reorder the terms for easier multiplication: 33 + -3n = -5 * 8n(9 + n) Multiply -5 * 8 33 + -3n = -40n(9 + n) 33 + -3n = (9 * -40n + n * -40n) 33 + -3n = (-360n + -40n2) Solving 33 + -3n = -360n + -40n2 Solving for variable 'n'. Combine like terms: -3n + 360n = 357n 33 + 357n + 40n2 = -360n + -40n2 + 360n + 40n2 Reorder the terms: 33 + 357n + 40n2 = -360n + 360n + -40n2 + 40n2 Combine like terms: -360n + 360n = 0 33 + 357n + 40n2 = 0 + -40n2 + 40n2 33 + 357n + 40n2 = -40n2 + 40n2 Combine like terms: -40n2 + 40n2 = 0 33 + 357n + 40n2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. 0.825 + 8.925n + n2 = 0 Move the constant term to the right: Add '-0.825' to each side of the equation. 0.825 + 8.925n + -0.825 + n2 = 0 + -0.825 Reorder the terms: 0.825 + -0.825 + 8.925n + n2 = 0 + -0.825 Combine like terms: 0.825 + -0.825 = 0.000 0.000 + 8.925n + n2 = 0 + -0.825 8.925n + n2 = 0 + -0.825 Combine like terms: 0 + -0.825 = -0.825 8.925n + n2 = -0.825 The n term is 8.925n. Take half its coefficient (4.4625). Square it (19.91390625) and add it to both sides. Add '19.91390625' to each side of the equation. 8.925n + 19.91390625 + n2 = -0.825 + 19.91390625 Reorder the terms: 19.91390625 + 8.925n + n2 = -0.825 + 19.91390625 Combine like terms: -0.825 + 19.91390625 = 19.08890625 19.91390625 + 8.925n + n2 = 19.08890625 Factor a perfect square on the left side: (n + 4.4625)(n + 4.4625) = 19.08890625 Calculate the square root of the right side: 4.369085288 Break this problem into two subproblems by setting (n + 4.4625) equal to 4.369085288 and -4.369085288.Subproblem 1
n + 4.4625 = 4.369085288 Simplifying n + 4.4625 = 4.369085288 Reorder the terms: 4.4625 + n = 4.369085288 Solving 4.4625 + n = 4.369085288 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.4625' to each side of the equation. 4.4625 + -4.4625 + n = 4.369085288 + -4.4625 Combine like terms: 4.4625 + -4.4625 = 0.0000 0.0000 + n = 4.369085288 + -4.4625 n = 4.369085288 + -4.4625 Combine like terms: 4.369085288 + -4.4625 = -0.093414712 n = -0.093414712 Simplifying n = -0.093414712Subproblem 2
n + 4.4625 = -4.369085288 Simplifying n + 4.4625 = -4.369085288 Reorder the terms: 4.4625 + n = -4.369085288 Solving 4.4625 + n = -4.369085288 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.4625' to each side of the equation. 4.4625 + -4.4625 + n = -4.369085288 + -4.4625 Combine like terms: 4.4625 + -4.4625 = 0.0000 0.0000 + n = -4.369085288 + -4.4625 n = -4.369085288 + -4.4625 Combine like terms: -4.369085288 + -4.4625 = -8.831585288 n = -8.831585288 Simplifying n = -8.831585288Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.093414712, -8.831585288}
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