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Simplifying 4n2 = -12n + -6 Reorder the terms: 4n2 = -6 + -12n Solving 4n2 = -6 + -12n Solving for variable 'n'. Reorder the terms: 6 + 12n + 4n2 = -6 + -12n + 6 + 12n Reorder the terms: 6 + 12n + 4n2 = -6 + 6 + -12n + 12n Combine like terms: -6 + 6 = 0 6 + 12n + 4n2 = 0 + -12n + 12n 6 + 12n + 4n2 = -12n + 12n Combine like terms: -12n + 12n = 0 6 + 12n + 4n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(3 + 6n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(3 + 6n + 2n2)' equal to zero and attempt to solve: Simplifying 3 + 6n + 2n2 = 0 Solving 3 + 6n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 3n + n2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3n + -1.5 + n2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3n + n2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3n + n2 = 0 + -1.5 3n + n2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3n + n2 = -1.5 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 = -1.5 + 2.25 Reorder the terms: 2.25 + 3n + n2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + 3n + n2 = 0.75 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (n + 1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
n + 1.5 = 0.866025404 Simplifying n + 1.5 = 0.866025404 Reorder the terms: 1.5 + n = 0.866025404 Solving 1.5 + n = 0.866025404 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 = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 0.866025404 + -1.5 n = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 n = -0.633974596 Simplifying n = -0.633974596Subproblem 2
n + 1.5 = -0.866025404 Simplifying n + 1.5 = -0.866025404 Reorder the terms: 1.5 + n = -0.866025404 Solving 1.5 + n = -0.866025404 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 = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -0.866025404 + -1.5 n = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 n = -2.366025404 Simplifying n = -2.366025404Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.633974596, -2.366025404}Solution
n = {-0.633974596, -2.366025404}
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