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Simplifying 3n2 + -71n + 24 = 0 Reorder the terms: 24 + -71n + 3n2 = 0 Solving 24 + -71n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 8 + -23.66666667n + n2 = 0 Move the constant term to the right: Add '-8' to each side of the equation. 8 + -23.66666667n + -8 + n2 = 0 + -8 Reorder the terms: 8 + -8 + -23.66666667n + n2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + -23.66666667n + n2 = 0 + -8 -23.66666667n + n2 = 0 + -8 Combine like terms: 0 + -8 = -8 -23.66666667n + n2 = -8 The n term is -23.66666667n. Take half its coefficient (-11.83333334). Square it (140.0277779) and add it to both sides. Add '140.0277779' to each side of the equation. -23.66666667n + 140.0277779 + n2 = -8 + 140.0277779 Reorder the terms: 140.0277779 + -23.66666667n + n2 = -8 + 140.0277779 Combine like terms: -8 + 140.0277779 = 132.0277779 140.0277779 + -23.66666667n + n2 = 132.0277779 Factor a perfect square on the left side: (n + -11.83333334)(n + -11.83333334) = 132.0277779 Calculate the square root of the right side: 11.490334107 Break this problem into two subproblems by setting (n + -11.83333334) equal to 11.490334107 and -11.490334107.Subproblem 1
n + -11.83333334 = 11.490334107 Simplifying n + -11.83333334 = 11.490334107 Reorder the terms: -11.83333334 + n = 11.490334107 Solving -11.83333334 + n = 11.490334107 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.83333334' to each side of the equation. -11.83333334 + 11.83333334 + n = 11.490334107 + 11.83333334 Combine like terms: -11.83333334 + 11.83333334 = 0.00000000 0.00000000 + n = 11.490334107 + 11.83333334 n = 11.490334107 + 11.83333334 Combine like terms: 11.490334107 + 11.83333334 = 23.323667447 n = 23.323667447 Simplifying n = 23.323667447Subproblem 2
n + -11.83333334 = -11.490334107 Simplifying n + -11.83333334 = -11.490334107 Reorder the terms: -11.83333334 + n = -11.490334107 Solving -11.83333334 + n = -11.490334107 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11.83333334' to each side of the equation. -11.83333334 + 11.83333334 + n = -11.490334107 + 11.83333334 Combine like terms: -11.83333334 + 11.83333334 = 0.00000000 0.00000000 + n = -11.490334107 + 11.83333334 n = -11.490334107 + 11.83333334 Combine like terms: -11.490334107 + 11.83333334 = 0.342999233 n = 0.342999233 Simplifying n = 0.342999233Solution
The solution to the problem is based on the solutions from the subproblems. n = {23.323667447, 0.342999233}
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