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Simplifying 6n2 = -12n + 11 Reorder the terms: 6n2 = 11 + -12n Solving 6n2 = 11 + -12n Solving for variable 'n'. Reorder the terms: -11 + 12n + 6n2 = 11 + -12n + -11 + 12n Reorder the terms: -11 + 12n + 6n2 = 11 + -11 + -12n + 12n Combine like terms: 11 + -11 = 0 -11 + 12n + 6n2 = 0 + -12n + 12n -11 + 12n + 6n2 = -12n + 12n Combine like terms: -12n + 12n = 0 -11 + 12n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.833333333 + 2n + n2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + 2n + 1.833333333 + n2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + 2n + n2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + 2n + n2 = 0 + 1.833333333 2n + n2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 2n + n2 = 1.833333333 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 = 1.833333333 + 1 Reorder the terms: 1 + 2n + n2 = 1.833333333 + 1 Combine like terms: 1.833333333 + 1 = 2.833333333 1 + 2n + n2 = 2.833333333 Factor a perfect square on the left side: (n + 1)(n + 1) = 2.833333333 Calculate the square root of the right side: 1.683250823 Break this problem into two subproblems by setting (n + 1) equal to 1.683250823 and -1.683250823.Subproblem 1
n + 1 = 1.683250823 Simplifying n + 1 = 1.683250823 Reorder the terms: 1 + n = 1.683250823 Solving 1 + n = 1.683250823 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 = 1.683250823 + -1 Combine like terms: 1 + -1 = 0 0 + n = 1.683250823 + -1 n = 1.683250823 + -1 Combine like terms: 1.683250823 + -1 = 0.683250823 n = 0.683250823 Simplifying n = 0.683250823Subproblem 2
n + 1 = -1.683250823 Simplifying n + 1 = -1.683250823 Reorder the terms: 1 + n = -1.683250823 Solving 1 + n = -1.683250823 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 = -1.683250823 + -1 Combine like terms: 1 + -1 = 0 0 + n = -1.683250823 + -1 n = -1.683250823 + -1 Combine like terms: -1.683250823 + -1 = -2.683250823 n = -2.683250823 Simplifying n = -2.683250823Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.683250823, -2.683250823}
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