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Simplifying 6n2 + 12n + 2 = 0 Reorder the terms: 2 + 12n + 6n2 = 0 Solving 2 + 12n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 6n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 6n + 3n2)' equal to zero and attempt to solve: Simplifying 1 + 6n + 3n2 = 0 Solving 1 + 6n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333 + 2n + n2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + 2n + -0.3333333333 + n2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + 2n + n2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + 2n + n2 = 0 + -0.3333333333 2n + n2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 2n + n2 = -0.3333333333 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 = -0.3333333333 + 1 Reorder the terms: 1 + 2n + n2 = -0.3333333333 + 1 Combine like terms: -0.3333333333 + 1 = 0.6666666667 1 + 2n + n2 = 0.6666666667 Factor a perfect square on the left side: (n + 1)(n + 1) = 0.6666666667 Calculate the square root of the right side: 0.816496581 Break this problem into two subproblems by setting (n + 1) equal to 0.816496581 and -0.816496581.Subproblem 1
n + 1 = 0.816496581 Simplifying n + 1 = 0.816496581 Reorder the terms: 1 + n = 0.816496581 Solving 1 + n = 0.816496581 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 = 0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + n = 0.816496581 + -1 n = 0.816496581 + -1 Combine like terms: 0.816496581 + -1 = -0.183503419 n = -0.183503419 Simplifying n = -0.183503419Subproblem 2
n + 1 = -0.816496581 Simplifying n + 1 = -0.816496581 Reorder the terms: 1 + n = -0.816496581 Solving 1 + n = -0.816496581 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 = -0.816496581 + -1 Combine like terms: 1 + -1 = 0 0 + n = -0.816496581 + -1 n = -0.816496581 + -1 Combine like terms: -0.816496581 + -1 = -1.816496581 n = -1.816496581 Simplifying n = -1.816496581Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.183503419, -1.816496581}Solution
n = {-0.183503419, -1.816496581}
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