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Simplifying 3n2 + 11n + -500 = 0 Reorder the terms: -500 + 11n + 3n2 = 0 Solving -500 + 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'. -166.6666667 + 3.666666667n + n2 = 0 Move the constant term to the right: Add '166.6666667' to each side of the equation. -166.6666667 + 3.666666667n + 166.6666667 + n2 = 0 + 166.6666667 Reorder the terms: -166.6666667 + 166.6666667 + 3.666666667n + n2 = 0 + 166.6666667 Combine like terms: -166.6666667 + 166.6666667 = 0.0000000 0.0000000 + 3.666666667n + n2 = 0 + 166.6666667 3.666666667n + n2 = 0 + 166.6666667 Combine like terms: 0 + 166.6666667 = 166.6666667 3.666666667n + n2 = 166.6666667 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 = 166.6666667 + 3.361111114 Reorder the terms: 3.361111114 + 3.666666667n + n2 = 166.6666667 + 3.361111114 Combine like terms: 166.6666667 + 3.361111114 = 170.027777814 3.361111114 + 3.666666667n + n2 = 170.027777814 Factor a perfect square on the left side: (n + 1.833333334)(n + 1.833333334) = 170.027777814 Calculate the square root of the right side: 13.039469997 Break this problem into two subproblems by setting (n + 1.833333334) equal to 13.039469997 and -13.039469997.Subproblem 1
n + 1.833333334 = 13.039469997 Simplifying n + 1.833333334 = 13.039469997 Reorder the terms: 1.833333334 + n = 13.039469997 Solving 1.833333334 + n = 13.039469997 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 = 13.039469997 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = 13.039469997 + -1.833333334 n = 13.039469997 + -1.833333334 Combine like terms: 13.039469997 + -1.833333334 = 11.206136663 n = 11.206136663 Simplifying n = 11.206136663Subproblem 2
n + 1.833333334 = -13.039469997 Simplifying n + 1.833333334 = -13.039469997 Reorder the terms: 1.833333334 + n = -13.039469997 Solving 1.833333334 + n = -13.039469997 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 = -13.039469997 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + n = -13.039469997 + -1.833333334 n = -13.039469997 + -1.833333334 Combine like terms: -13.039469997 + -1.833333334 = -14.872803331 n = -14.872803331 Simplifying n = -14.872803331Solution
The solution to the problem is based on the solutions from the subproblems. n = {11.206136663, -14.872803331}
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