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Simplifying 6n2 + 9 + 12n + -6 + -24 = 0 Reorder the terms: 9 + -6 + -24 + 12n + 6n2 = 0 Combine like terms: 9 + -6 = 3 3 + -24 + 12n + 6n2 = 0 Combine like terms: 3 + -24 = -21 -21 + 12n + 6n2 = 0 Solving -21 + 12n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-7 + 4n + 2n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-7 + 4n + 2n2)' equal to zero and attempt to solve: Simplifying -7 + 4n + 2n2 = 0 Solving -7 + 4n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 2n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 2n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 2n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 2n + n2 = 0 + 3.5 2n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 2n + n2 = 3.5 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 = 3.5 + 1 Reorder the terms: 1 + 2n + n2 = 3.5 + 1 Combine like terms: 3.5 + 1 = 4.5 1 + 2n + n2 = 4.5 Factor a perfect square on the left side: (n + 1)(n + 1) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (n + 1) equal to 2.121320344 and -2.121320344.Subproblem 1
n + 1 = 2.121320344 Simplifying n + 1 = 2.121320344 Reorder the terms: 1 + n = 2.121320344 Solving 1 + n = 2.121320344 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 = 2.121320344 + -1 Combine like terms: 1 + -1 = 0 0 + n = 2.121320344 + -1 n = 2.121320344 + -1 Combine like terms: 2.121320344 + -1 = 1.121320344 n = 1.121320344 Simplifying n = 1.121320344Subproblem 2
n + 1 = -2.121320344 Simplifying n + 1 = -2.121320344 Reorder the terms: 1 + n = -2.121320344 Solving 1 + n = -2.121320344 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 = -2.121320344 + -1 Combine like terms: 1 + -1 = 0 0 + n = -2.121320344 + -1 n = -2.121320344 + -1 Combine like terms: -2.121320344 + -1 = -3.121320344 n = -3.121320344 Simplifying n = -3.121320344Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.121320344, -3.121320344}Solution
n = {1.121320344, -3.121320344}
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