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Simplifying 6n2 = -12n + -5 Reorder the terms: 6n2 = -5 + -12n Solving 6n2 = -5 + -12n Solving for variable 'n'. Reorder the terms: 5 + 12n + 6n2 = -5 + -12n + 5 + 12n Reorder the terms: 5 + 12n + 6n2 = -5 + 5 + -12n + 12n Combine like terms: -5 + 5 = 0 5 + 12n + 6n2 = 0 + -12n + 12n 5 + 12n + 6n2 = -12n + 12n Combine like terms: -12n + 12n = 0 5 + 12n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.8333333333 + 2n + n2 = 0 Move the constant term to the right: Add '-0.8333333333' to each side of the equation. 0.8333333333 + 2n + -0.8333333333 + n2 = 0 + -0.8333333333 Reorder the terms: 0.8333333333 + -0.8333333333 + 2n + n2 = 0 + -0.8333333333 Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000 0.0000000000 + 2n + n2 = 0 + -0.8333333333 2n + n2 = 0 + -0.8333333333 Combine like terms: 0 + -0.8333333333 = -0.8333333333 2n + n2 = -0.8333333333 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 = -0.8333333333 + 1 Reorder the terms: 1 + 2n + n2 = -0.8333333333 + 1 Combine like terms: -0.8333333333 + 1 = 0.1666666667 1 + 2n + n2 = 0.1666666667 Factor a perfect square on the left side: (n + 1)(n + 1) = 0.1666666667 Calculate the square root of the right side: 0.408248291 Break this problem into two subproblems by setting (n + 1) equal to 0.408248291 and -0.408248291.Subproblem 1
n + 1 = 0.408248291 Simplifying n + 1 = 0.408248291 Reorder the terms: 1 + n = 0.408248291 Solving 1 + n = 0.408248291 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 = 0.408248291 + -1 Combine like terms: 1 + -1 = 0 0 + n = 0.408248291 + -1 n = 0.408248291 + -1 Combine like terms: 0.408248291 + -1 = -0.591751709 n = -0.591751709 Simplifying n = -0.591751709Subproblem 2
n + 1 = -0.408248291 Simplifying n + 1 = -0.408248291 Reorder the terms: 1 + n = -0.408248291 Solving 1 + n = -0.408248291 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 = -0.408248291 + -1 Combine like terms: 1 + -1 = 0 0 + n = -0.408248291 + -1 n = -0.408248291 + -1 Combine like terms: -0.408248291 + -1 = -1.408248291 n = -1.408248291 Simplifying n = -1.408248291Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.591751709, -1.408248291}
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