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Simplifying -9n2 + -24n + 16 = 0 Reorder the terms: 16 + -24n + -9n2 = 0 Solving 16 + -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'. -1.777777778 + 2.666666667n + n2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 2.666666667n + 1.777777778 + n2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 2.666666667n + n2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 2.666666667n + n2 = 0 + 1.777777778 2.666666667n + n2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 2.666666667n + n2 = 1.777777778 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 = 1.777777778 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667n + n2 = 1.777777778 + 1.777777780 Combine like terms: 1.777777778 + 1.777777780 = 3.555555558 1.777777780 + 2.666666667n + n2 = 3.555555558 Factor a perfect square on the left side: (n + 1.333333334)(n + 1.333333334) = 3.555555558 Calculate the square root of the right side: 1.885618084 Break this problem into two subproblems by setting (n + 1.333333334) equal to 1.885618084 and -1.885618084.Subproblem 1
n + 1.333333334 = 1.885618084 Simplifying n + 1.333333334 = 1.885618084 Reorder the terms: 1.333333334 + n = 1.885618084 Solving 1.333333334 + n = 1.885618084 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 = 1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = 1.885618084 + -1.333333334 n = 1.885618084 + -1.333333334 Combine like terms: 1.885618084 + -1.333333334 = 0.55228475 n = 0.55228475 Simplifying n = 0.55228475Subproblem 2
n + 1.333333334 = -1.885618084 Simplifying n + 1.333333334 = -1.885618084 Reorder the terms: 1.333333334 + n = -1.885618084 Solving 1.333333334 + n = -1.885618084 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 = -1.885618084 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + n = -1.885618084 + -1.333333334 n = -1.885618084 + -1.333333334 Combine like terms: -1.885618084 + -1.333333334 = -3.218951418 n = -3.218951418 Simplifying n = -3.218951418Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.55228475, -3.218951418}
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