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Simplifying 9n2 + 24n + 14 = 0 Reorder the terms: 14 + 24n + 9n2 = 0 Solving 14 + 24n + 9n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 1.555555556 + 2.666666667n + n2 = 0 Move the constant term to the right: Add '-1.555555556' to each side of the equation. 1.555555556 + 2.666666667n + -1.555555556 + n2 = 0 + -1.555555556 Reorder the terms: 1.555555556 + -1.555555556 + 2.666666667n + n2 = 0 + -1.555555556 Combine like terms: 1.555555556 + -1.555555556 = 0.000000000 0.000000000 + 2.666666667n + n2 = 0 + -1.555555556 2.666666667n + n2 = 0 + -1.555555556 Combine like terms: 0 + -1.555555556 = -1.555555556 2.666666667n + n2 = -1.555555556 The n term is 2.666666667n. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667n + 1.777777780 + n2 = -1.555555556 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667n + n2 = -1.555555556 + 1.777777780 Combine like terms: -1.555555556 + 1.777777780 = 0.222222224 1.777777780 + 2.666666667n + n2 = 0.222222224 Factor a perfect square on the left side: (n + 1.333333334)(n + 1.333333334) = 0.222222224 Calculate the square root of the right side: 0.471404523 Break this problem into two subproblems by setting (n + 1.333333334) equal to 0.471404523 and -0.471404523.Subproblem 1
n + 1.333333334 = 0.471404523 Simplifying n + 1.333333334 = 0.471404523 Reorder the terms: 1.333333334 + n = 0.471404523 Solving 1.333333334 + n = 0.471404523 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = 0.471404523 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 0.471404523 + -1.333333334 n = 0.471404523 + -1.333333334 Combine like terms: 0.471404523 + -1.333333334 = -0.861928811 n = -0.861928811 Simplifying n = -0.861928811Subproblem 2
n + 1.333333334 = -0.471404523 Simplifying n + 1.333333334 = -0.471404523 Reorder the terms: 1.333333334 + n = -0.471404523 Solving 1.333333334 + n = -0.471404523 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + n = -0.471404523 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -0.471404523 + -1.333333334 n = -0.471404523 + -1.333333334 Combine like terms: -0.471404523 + -1.333333334 = -1.804737857 n = -1.804737857 Simplifying n = -1.804737857Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.861928811, -1.804737857}
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