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Simplifying 2n2 = -12n + -7 Reorder the terms: 2n2 = -7 + -12n Solving 2n2 = -7 + -12n Solving for variable 'n'. Reorder the terms: 7 + 12n + 2n2 = -7 + -12n + 7 + 12n Reorder the terms: 7 + 12n + 2n2 = -7 + 7 + -12n + 12n Combine like terms: -7 + 7 = 0 7 + 12n + 2n2 = 0 + -12n + 12n 7 + 12n + 2n2 = -12n + 12n Combine like terms: -12n + 12n = 0 7 + 12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + 6n + n2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + 6n + -3.5 + n2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + 6n + n2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + 6n + n2 = 0 + -3.5 6n + n2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 6n + n2 = -3.5 The n term is 6n. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6n + 9 + n2 = -3.5 + 9 Reorder the terms: 9 + 6n + n2 = -3.5 + 9 Combine like terms: -3.5 + 9 = 5.5 9 + 6n + n2 = 5.5 Factor a perfect square on the left side: (n + 3)(n + 3) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (n + 3) equal to 2.34520788 and -2.34520788.Subproblem 1
n + 3 = 2.34520788 Simplifying n + 3 = 2.34520788 Reorder the terms: 3 + n = 2.34520788 Solving 3 + n = 2.34520788 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 2.34520788 + -3 Combine like terms: 3 + -3 = 0 0 + n = 2.34520788 + -3 n = 2.34520788 + -3 Combine like terms: 2.34520788 + -3 = -0.65479212 n = -0.65479212 Simplifying n = -0.65479212Subproblem 2
n + 3 = -2.34520788 Simplifying n + 3 = -2.34520788 Reorder the terms: 3 + n = -2.34520788 Solving 3 + n = -2.34520788 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -2.34520788 + -3 Combine like terms: 3 + -3 = 0 0 + n = -2.34520788 + -3 n = -2.34520788 + -3 Combine like terms: -2.34520788 + -3 = -5.34520788 n = -5.34520788 Simplifying n = -5.34520788Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.65479212, -5.34520788}
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