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Simplifying 5n2 + 19 = 3n + 92 + -3n2 Reorder the terms: 19 + 5n2 = 3n + 92 + -3n2 Reorder the terms: 19 + 5n2 = 92 + 3n + -3n2 Solving 19 + 5n2 = 92 + 3n + -3n2 Solving for variable 'n'. Reorder the terms: 19 + -92 + -3n + 5n2 + 3n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Combine like terms: 19 + -92 = -73 -73 + -3n + 5n2 + 3n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Combine like terms: 5n2 + 3n2 = 8n2 -73 + -3n + 8n2 = 92 + 3n + -3n2 + -92 + -3n + 3n2 Reorder the terms: -73 + -3n + 8n2 = 92 + -92 + 3n + -3n + -3n2 + 3n2 Combine like terms: 92 + -92 = 0 -73 + -3n + 8n2 = 0 + 3n + -3n + -3n2 + 3n2 -73 + -3n + 8n2 = 3n + -3n + -3n2 + 3n2 Combine like terms: 3n + -3n = 0 -73 + -3n + 8n2 = 0 + -3n2 + 3n2 -73 + -3n + 8n2 = -3n2 + 3n2 Combine like terms: -3n2 + 3n2 = 0 -73 + -3n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -9.125 + -0.375n + n2 = 0 Move the constant term to the right: Add '9.125' to each side of the equation. -9.125 + -0.375n + 9.125 + n2 = 0 + 9.125 Reorder the terms: -9.125 + 9.125 + -0.375n + n2 = 0 + 9.125 Combine like terms: -9.125 + 9.125 = 0.000 0.000 + -0.375n + n2 = 0 + 9.125 -0.375n + n2 = 0 + 9.125 Combine like terms: 0 + 9.125 = 9.125 -0.375n + n2 = 9.125 The n term is -0.375n. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375n + 0.03515625 + n2 = 9.125 + 0.03515625 Reorder the terms: 0.03515625 + -0.375n + n2 = 9.125 + 0.03515625 Combine like terms: 9.125 + 0.03515625 = 9.16015625 0.03515625 + -0.375n + n2 = 9.16015625 Factor a perfect square on the left side: (n + -0.1875)(n + -0.1875) = 9.16015625 Calculate the square root of the right side: 3.026575003 Break this problem into two subproblems by setting (n + -0.1875) equal to 3.026575003 and -3.026575003.Subproblem 1
n + -0.1875 = 3.026575003 Simplifying n + -0.1875 = 3.026575003 Reorder the terms: -0.1875 + n = 3.026575003 Solving -0.1875 + n = 3.026575003 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = 3.026575003 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = 3.026575003 + 0.1875 n = 3.026575003 + 0.1875 Combine like terms: 3.026575003 + 0.1875 = 3.214075003 n = 3.214075003 Simplifying n = 3.214075003Subproblem 2
n + -0.1875 = -3.026575003 Simplifying n + -0.1875 = -3.026575003 Reorder the terms: -0.1875 + n = -3.026575003 Solving -0.1875 + n = -3.026575003 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + n = -3.026575003 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + n = -3.026575003 + 0.1875 n = -3.026575003 + 0.1875 Combine like terms: -3.026575003 + 0.1875 = -2.839075003 n = -2.839075003 Simplifying n = -2.839075003Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.214075003, -2.839075003}
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