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Simplifying 542n2 + -678n + -456 = 542 Reorder the terms: -456 + -678n + 542n2 = 542 Solving -456 + -678n + 542n2 = 542 Solving for variable 'n'. Reorder the terms: -456 + -542 + -678n + 542n2 = 542 + -542 Combine like terms: -456 + -542 = -998 -998 + -678n + 542n2 = 542 + -542 Combine like terms: 542 + -542 = 0 -998 + -678n + 542n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-499 + -339n + 271n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-499 + -339n + 271n2)' equal to zero and attempt to solve: Simplifying -499 + -339n + 271n2 = 0 Solving -499 + -339n + 271n2 = 0 Begin completing the square. Divide all terms by 271 the coefficient of the squared term: Divide each side by '271'. -1.841328413 + -1.250922509n + n2 = 0 Move the constant term to the right: Add '1.841328413' to each side of the equation. -1.841328413 + -1.250922509n + 1.841328413 + n2 = 0 + 1.841328413 Reorder the terms: -1.841328413 + 1.841328413 + -1.250922509n + n2 = 0 + 1.841328413 Combine like terms: -1.841328413 + 1.841328413 = 0.000000000 0.000000000 + -1.250922509n + n2 = 0 + 1.841328413 -1.250922509n + n2 = 0 + 1.841328413 Combine like terms: 0 + 1.841328413 = 1.841328413 -1.250922509n + n2 = 1.841328413 The n term is -1.250922509n. Take half its coefficient (-0.6254612545). Square it (0.3912017809) and add it to both sides. Add '0.3912017809' to each side of the equation. -1.250922509n + 0.3912017809 + n2 = 1.841328413 + 0.3912017809 Reorder the terms: 0.3912017809 + -1.250922509n + n2 = 1.841328413 + 0.3912017809 Combine like terms: 1.841328413 + 0.3912017809 = 2.2325301939 0.3912017809 + -1.250922509n + n2 = 2.2325301939 Factor a perfect square on the left side: (n + -0.6254612545)(n + -0.6254612545) = 2.2325301939 Calculate the square root of the right side: 1.494165384 Break this problem into two subproblems by setting (n + -0.6254612545) equal to 1.494165384 and -1.494165384.Subproblem 1
n + -0.6254612545 = 1.494165384 Simplifying n + -0.6254612545 = 1.494165384 Reorder the terms: -0.6254612545 + n = 1.494165384 Solving -0.6254612545 + n = 1.494165384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6254612545' to each side of the equation. -0.6254612545 + 0.6254612545 + n = 1.494165384 + 0.6254612545 Combine like terms: -0.6254612545 + 0.6254612545 = 0.0000000000 0.0000000000 + n = 1.494165384 + 0.6254612545 n = 1.494165384 + 0.6254612545 Combine like terms: 1.494165384 + 0.6254612545 = 2.1196266385 n = 2.1196266385 Simplifying n = 2.1196266385Subproblem 2
n + -0.6254612545 = -1.494165384 Simplifying n + -0.6254612545 = -1.494165384 Reorder the terms: -0.6254612545 + n = -1.494165384 Solving -0.6254612545 + n = -1.494165384 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6254612545' to each side of the equation. -0.6254612545 + 0.6254612545 + n = -1.494165384 + 0.6254612545 Combine like terms: -0.6254612545 + 0.6254612545 = 0.0000000000 0.0000000000 + n = -1.494165384 + 0.6254612545 n = -1.494165384 + 0.6254612545 Combine like terms: -1.494165384 + 0.6254612545 = -0.8687041295 n = -0.8687041295 Simplifying n = -0.8687041295Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.1196266385, -0.8687041295}Solution
n = {2.1196266385, -0.8687041295}
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