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Simplifying 6n2 = 21 + -2n Solving 6n2 = 21 + -2n Solving for variable 'n'. Reorder the terms: -21 + 2n + 6n2 = 21 + -2n + -21 + 2n Reorder the terms: -21 + 2n + 6n2 = 21 + -21 + -2n + 2n Combine like terms: 21 + -21 = 0 -21 + 2n + 6n2 = 0 + -2n + 2n -21 + 2n + 6n2 = -2n + 2n Combine like terms: -2n + 2n = 0 -21 + 2n + 6n2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.5 + 0.3333333333n + n2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 0.3333333333n + 3.5 + n2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 0.3333333333n + n2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 0.3333333333n + n2 = 0 + 3.5 0.3333333333n + n2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 0.3333333333n + n2 = 3.5 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 = 3.5 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333n + n2 = 3.5 + 0.02777777779 Combine like terms: 3.5 + 0.02777777779 = 3.52777777779 0.02777777779 + 0.3333333333n + n2 = 3.52777777779 Factor a perfect square on the left side: (n + 0.1666666667)(n + 0.1666666667) = 3.52777777779 Calculate the square root of the right side: 1.878237945 Break this problem into two subproblems by setting (n + 0.1666666667) equal to 1.878237945 and -1.878237945.Subproblem 1
n + 0.1666666667 = 1.878237945 Simplifying n + 0.1666666667 = 1.878237945 Reorder the terms: 0.1666666667 + n = 1.878237945 Solving 0.1666666667 + n = 1.878237945 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 = 1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + n = 1.878237945 + -0.1666666667 n = 1.878237945 + -0.1666666667 Combine like terms: 1.878237945 + -0.1666666667 = 1.7115712783 n = 1.7115712783 Simplifying n = 1.7115712783Subproblem 2
n + 0.1666666667 = -1.878237945 Simplifying n + 0.1666666667 = -1.878237945 Reorder the terms: 0.1666666667 + n = -1.878237945 Solving 0.1666666667 + n = -1.878237945 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 = -1.878237945 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + n = -1.878237945 + -0.1666666667 n = -1.878237945 + -0.1666666667 Combine like terms: -1.878237945 + -0.1666666667 = -2.0449046117 n = -2.0449046117 Simplifying n = -2.0449046117Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.7115712783, -2.0449046117}
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