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Simplifying 3n2 + 65n + 24 = 0 Reorder the terms: 24 + 65n + 3n2 = 0 Solving 24 + 65n + 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 + 21.66666667n + n2 = 0 Move the constant term to the right: Add '-8' to each side of the equation. 8 + 21.66666667n + -8 + n2 = 0 + -8 Reorder the terms: 8 + -8 + 21.66666667n + n2 = 0 + -8 Combine like terms: 8 + -8 = 0 0 + 21.66666667n + n2 = 0 + -8 21.66666667n + n2 = 0 + -8 Combine like terms: 0 + -8 = -8 21.66666667n + n2 = -8 The n term is 21.66666667n. Take half its coefficient (10.83333334). Square it (117.3611113) and add it to both sides. Add '117.3611113' to each side of the equation. 21.66666667n + 117.3611113 + n2 = -8 + 117.3611113 Reorder the terms: 117.3611113 + 21.66666667n + n2 = -8 + 117.3611113 Combine like terms: -8 + 117.3611113 = 109.3611113 117.3611113 + 21.66666667n + n2 = 109.3611113 Factor a perfect square on the left side: (n + 10.83333334)(n + 10.83333334) = 109.3611113 Calculate the square root of the right side: 10.457586304 Break this problem into two subproblems by setting (n + 10.83333334) equal to 10.457586304 and -10.457586304.Subproblem 1
n + 10.83333334 = 10.457586304 Simplifying n + 10.83333334 = 10.457586304 Reorder the terms: 10.83333334 + n = 10.457586304 Solving 10.83333334 + n = 10.457586304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.83333334' to each side of the equation. 10.83333334 + -10.83333334 + n = 10.457586304 + -10.83333334 Combine like terms: 10.83333334 + -10.83333334 = 0.00000000 0.00000000 + n = 10.457586304 + -10.83333334 n = 10.457586304 + -10.83333334 Combine like terms: 10.457586304 + -10.83333334 = -0.375747036 n = -0.375747036 Simplifying n = -0.375747036Subproblem 2
n + 10.83333334 = -10.457586304 Simplifying n + 10.83333334 = -10.457586304 Reorder the terms: 10.83333334 + n = -10.457586304 Solving 10.83333334 + n = -10.457586304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.83333334' to each side of the equation. 10.83333334 + -10.83333334 + n = -10.457586304 + -10.83333334 Combine like terms: 10.83333334 + -10.83333334 = 0.00000000 0.00000000 + n = -10.457586304 + -10.83333334 n = -10.457586304 + -10.83333334 Combine like terms: -10.457586304 + -10.83333334 = -21.290919644 n = -21.290919644 Simplifying n = -21.290919644Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.375747036, -21.290919644}
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