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Simplifying 0 = 9n2 + -3n + -968 Reorder the terms: 0 = -968 + -3n + 9n2 Solving 0 = -968 + -3n + 9n2 Solving for variable 'n'. Combine like terms: 0 + 968 = 968 968 + 3n + -9n2 = -968 + -3n + 9n2 + 968 + 3n + -9n2 Reorder the terms: 968 + 3n + -9n2 = -968 + 968 + -3n + 3n + 9n2 + -9n2 Combine like terms: -968 + 968 = 0 968 + 3n + -9n2 = 0 + -3n + 3n + 9n2 + -9n2 968 + 3n + -9n2 = -3n + 3n + 9n2 + -9n2 Combine like terms: -3n + 3n = 0 968 + 3n + -9n2 = 0 + 9n2 + -9n2 968 + 3n + -9n2 = 9n2 + -9n2 Combine like terms: 9n2 + -9n2 = 0 968 + 3n + -9n2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -107.5555556 + -0.3333333333n + n2 = 0 Move the constant term to the right: Add '107.5555556' to each side of the equation. -107.5555556 + -0.3333333333n + 107.5555556 + n2 = 0 + 107.5555556 Reorder the terms: -107.5555556 + 107.5555556 + -0.3333333333n + n2 = 0 + 107.5555556 Combine like terms: -107.5555556 + 107.5555556 = 0.0000000 0.0000000 + -0.3333333333n + n2 = 0 + 107.5555556 -0.3333333333n + n2 = 0 + 107.5555556 Combine like terms: 0 + 107.5555556 = 107.5555556 -0.3333333333n + n2 = 107.5555556 The n term is -0.3333333333n. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333n + 0.02777777779 + n2 = 107.5555556 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333n + n2 = 107.5555556 + 0.02777777779 Combine like terms: 107.5555556 + 0.02777777779 = 107.58333337779 0.02777777779 + -0.3333333333n + n2 = 107.58333337779 Factor a perfect square on the left side: (n + -0.1666666667)(n + -0.1666666667) = 107.58333337779 Calculate the square root of the right side: 10.37223859 Break this problem into two subproblems by setting (n + -0.1666666667) equal to 10.37223859 and -10.37223859.Subproblem 1
n + -0.1666666667 = 10.37223859 Simplifying n + -0.1666666667 = 10.37223859 Reorder the terms: -0.1666666667 + n = 10.37223859 Solving -0.1666666667 + n = 10.37223859 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = 10.37223859 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = 10.37223859 + 0.1666666667 n = 10.37223859 + 0.1666666667 Combine like terms: 10.37223859 + 0.1666666667 = 10.5389052567 n = 10.5389052567 Simplifying n = 10.5389052567Subproblem 2
n + -0.1666666667 = -10.37223859 Simplifying n + -0.1666666667 = -10.37223859 Reorder the terms: -0.1666666667 + n = -10.37223859 Solving -0.1666666667 + n = -10.37223859 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = -10.37223859 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = -10.37223859 + 0.1666666667 n = -10.37223859 + 0.1666666667 Combine like terms: -10.37223859 + 0.1666666667 = -10.2055719233 n = -10.2055719233 Simplifying n = -10.2055719233Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.5389052567, -10.2055719233}
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