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Simplifying 5n2 + 19n = 3n + 92 + -3n2 Reorder the terms: 19n + 5n2 = 3n + 92 + -3n2 Reorder the terms: 19n + 5n2 = 92 + 3n + -3n2 Solving 19n + 5n2 = 92 + 3n + -3n2 Solving for variable 'n'. Reorder the terms: -92 + 19n + -3n + 5n2 + 3n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Combine like terms: 19n + -3n = 16n -92 + 16n + 5n2 + 3n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Combine like terms: 5n2 + 3n2 = 8n2 -92 + 16n + 8n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Reorder the terms: -92 + 16n + 8n2 = 92 + -92 + 3n + -3n + -3n2 + 3n2 Combine like terms: 92 + -92 = 0 -92 + 16n + 8n2 = 0 + 3n + -3n + -3n2 + 3n2 -92 + 16n + 8n2 = 3n + -3n + -3n2 + 3n2 Combine like terms: 3n + -3n = 0 -92 + 16n + 8n2 = 0 + -3n2 + 3n2 -92 + 16n + 8n2 = -3n2 + 3n2 Combine like terms: -3n2 + 3n2 = 0 -92 + 16n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-23 + 4n + 2n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-23 + 4n + 2n2)' equal to zero and attempt to solve: Simplifying -23 + 4n + 2n2 = 0 Solving -23 + 4n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11.5 + 2n + n2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 2n + 11.5 + n2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 2n + n2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 2n + n2 = 0 + 11.5 2n + n2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 2n + n2 = 11.5 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 11.5 + 1 Reorder the terms: 1 + 2n + n2 = 11.5 + 1 Combine like terms: 11.5 + 1 = 12.5 1 + 2n + n2 = 12.5 Factor a perfect square on the left side: (n + 1)(n + 1) = 12.5 Calculate the square root of the right side: 3.535533906 Break this problem into two subproblems by setting (n + 1) equal to 3.535533906 and -3.535533906.Subproblem 1
n + 1 = 3.535533906 Simplifying n + 1 = 3.535533906 Reorder the terms: 1 + n = 3.535533906 Solving 1 + n = 3.535533906 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 3.535533906 + -1 Combine like terms: 1 + -1 = 0 0 + n = 3.535533906 + -1 n = 3.535533906 + -1 Combine like terms: 3.535533906 + -1 = 2.535533906 n = 2.535533906 Simplifying n = 2.535533906Subproblem 2
n + 1 = -3.535533906 Simplifying n + 1 = -3.535533906 Reorder the terms: 1 + n = -3.535533906 Solving 1 + n = -3.535533906 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -3.535533906 + -1 Combine like terms: 1 + -1 = 0 0 + n = -3.535533906 + -1 n = -3.535533906 + -1 Combine like terms: -3.535533906 + -1 = -4.535533906 n = -4.535533906 Simplifying n = -4.535533906Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.535533906, -4.535533906}Solution
n = {2.535533906, -4.535533906}
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