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Simplifying 4n2 + 12n = 210 Reorder the terms: 12n + 4n2 = 210 Solving 12n + 4n2 = 210 Solving for variable 'n'. Reorder the terms: -210 + 12n + 4n2 = 210 + -210 Combine like terms: 210 + -210 = 0 -210 + 12n + 4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-105 + 6n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-105 + 6n + 2n2)' equal to zero and attempt to solve: Simplifying -105 + 6n + 2n2 = 0 Solving -105 + 6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -52.5 + 3n + n2 = 0 Move the constant term to the right: Add '52.5' to each side of the equation. -52.5 + 3n + 52.5 + n2 = 0 + 52.5 Reorder the terms: -52.5 + 52.5 + 3n + n2 = 0 + 52.5 Combine like terms: -52.5 + 52.5 = 0.0 0.0 + 3n + n2 = 0 + 52.5 3n + n2 = 0 + 52.5 Combine like terms: 0 + 52.5 = 52.5 3n + n2 = 52.5 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 52.5 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 52.5 + 2.25 Combine like terms: 52.5 + 2.25 = 54.75 2.25 + 3n + n2 = 54.75 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 54.75 Calculate the square root of the right side: 7.399324293 Break this problem into two subproblems by setting (n + 1.5) equal to 7.399324293 and -7.399324293.Subproblem 1
n + 1.5 = 7.399324293 Simplifying n + 1.5 = 7.399324293 Reorder the terms: 1.5 + n = 7.399324293 Solving 1.5 + n = 7.399324293 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 7.399324293 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 7.399324293 + -1.5 n = 7.399324293 + -1.5 Combine like terms: 7.399324293 + -1.5 = 5.899324293 n = 5.899324293 Simplifying n = 5.899324293Subproblem 2
n + 1.5 = -7.399324293 Simplifying n + 1.5 = -7.399324293 Reorder the terms: 1.5 + n = -7.399324293 Solving 1.5 + n = -7.399324293 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -7.399324293 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -7.399324293 + -1.5 n = -7.399324293 + -1.5 Combine like terms: -7.399324293 + -1.5 = -8.899324293 n = -8.899324293 Simplifying n = -8.899324293Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.899324293, -8.899324293}Solution
n = {5.899324293, -8.899324293}
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