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Simplifying 3n2 + 11n + -6 = 0 Reorder the terms: -6 + 11n + 3n2 = 0 Solving -6 + 11n + 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'. -2 + 3.666666667n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 3.666666667n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 3.666666667n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 3.666666667n + n2 = 0 + 2 3.666666667n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 3.666666667n + n2 = 2 The n term is 3.666666667n. Take half its coefficient (1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. 3.666666667n + 3.361111114 + n2 = 2 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667n + n2 = 2 + 3.361111114 Combine like terms: 2 + 3.361111114 = 5.361111114 3.361111114 + 3.666666667n + n2 = 5.361111114 Factor a perfect square on the left side: (n + 1.833333334)(n + 1.833333334) = 5.361111114 Calculate the square root of the right side: 2.315407332 Break this problem into two subproblems by setting (n + 1.833333334) equal to 2.315407332 and -2.315407332.Subproblem 1
n + 1.833333334 = 2.315407332 Simplifying n + 1.833333334 = 2.315407332 Reorder the terms: 1.833333334 + n = 2.315407332 Solving 1.833333334 + n = 2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = 2.315407332 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = 2.315407332 + -1.833333334 n = 2.315407332 + -1.833333334 Combine like terms: 2.315407332 + -1.833333334 = 0.482073998 n = 0.482073998 Simplifying n = 0.482073998Subproblem 2
n + 1.833333334 = -2.315407332 Simplifying n + 1.833333334 = -2.315407332 Reorder the terms: 1.833333334 + n = -2.315407332 Solving 1.833333334 + n = -2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.833333334' to each side of the equation. 1.833333334 + -1.833333334 + n = -2.315407332 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = -2.315407332 + -1.833333334 n = -2.315407332 + -1.833333334 Combine like terms: -2.315407332 + -1.833333334 = -4.148740666 n = -4.148740666 Simplifying n = -4.148740666Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.482073998, -4.148740666}
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