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Simplifying -3n2 + 34n = 8n2 + 12 Reorder the terms: 34n + -3n2 = 8n2 + 12 Reorder the terms: 34n + -3n2 = 12 + 8n2 Solving 34n + -3n2 = 12 + 8n2 Solving for variable 'n'. Reorder the terms: -12 + 34n + -3n2 + -8n2 = 12 + 8n2 + -12 + -8n2 Combine like terms: -3n2 + -8n2 = -11n2 -12 + 34n + -11n2 = 12 + 8n2 + -12 + -8n2 Reorder the terms: -12 + 34n + -11n2 = 12 + -12 + 8n2 + -8n2 Combine like terms: 12 + -12 = 0 -12 + 34n + -11n2 = 0 + 8n2 + -8n2 -12 + 34n + -11n2 = 8n2 + -8n2 Combine like terms: 8n2 + -8n2 = 0 -12 + 34n + -11n2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. 1.090909091 + -3.090909091n + n2 = 0 Move the constant term to the right: Add '-1.090909091' to each side of the equation. 1.090909091 + -3.090909091n + -1.090909091 + n2 = 0 + -1.090909091 Reorder the terms: 1.090909091 + -1.090909091 + -3.090909091n + n2 = 0 + -1.090909091 Combine like terms: 1.090909091 + -1.090909091 = 0.000000000 0.000000000 + -3.090909091n + n2 = 0 + -1.090909091 -3.090909091n + n2 = 0 + -1.090909091 Combine like terms: 0 + -1.090909091 = -1.090909091 -3.090909091n + n2 = -1.090909091 The n term is -3.090909091n. Take half its coefficient (-1.545454546). Square it (2.388429754) and add it to both sides. Add '2.388429754' to each side of the equation. -3.090909091n + 2.388429754 + n2 = -1.090909091 + 2.388429754 Reorder the terms: 2.388429754 + -3.090909091n + n2 = -1.090909091 + 2.388429754 Combine like terms: -1.090909091 + 2.388429754 = 1.297520663 2.388429754 + -3.090909091n + n2 = 1.297520663 Factor a perfect square on the left side: (n + -1.545454546)(n + -1.545454546) = 1.297520663 Calculate the square root of the right side: 1.139087645 Break this problem into two subproblems by setting (n + -1.545454546) equal to 1.139087645 and -1.139087645.Subproblem 1
n + -1.545454546 = 1.139087645 Simplifying n + -1.545454546 = 1.139087645 Reorder the terms: -1.545454546 + n = 1.139087645 Solving -1.545454546 + n = 1.139087645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.545454546' to each side of the equation. -1.545454546 + 1.545454546 + n = 1.139087645 + 1.545454546 Combine like terms: -1.545454546 + 1.545454546 = 0.000000000 0.000000000 + n = 1.139087645 + 1.545454546 n = 1.139087645 + 1.545454546 Combine like terms: 1.139087645 + 1.545454546 = 2.684542191 n = 2.684542191 Simplifying n = 2.684542191Subproblem 2
n + -1.545454546 = -1.139087645 Simplifying n + -1.545454546 = -1.139087645 Reorder the terms: -1.545454546 + n = -1.139087645 Solving -1.545454546 + n = -1.139087645 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.545454546' to each side of the equation. -1.545454546 + 1.545454546 + n = -1.139087645 + 1.545454546 Combine like terms: -1.545454546 + 1.545454546 = 0.000000000 0.000000000 + n = -1.139087645 + 1.545454546 n = -1.139087645 + 1.545454546 Combine like terms: -1.139087645 + 1.545454546 = 0.406366901 n = 0.406366901 Simplifying n = 0.406366901Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.684542191, 0.406366901}
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