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Simplifying 3n2 + 165n + -500 = 0 Reorder the terms: -500 + 165n + 3n2 = 0 Solving -500 + 165n + 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 + 55n + n2 = 0 Move the constant term to the right: Add '166.6666667' to each side of the equation. -166.6666667 + 55n + 166.6666667 + n2 = 0 + 166.6666667 Reorder the terms: -166.6666667 + 166.6666667 + 55n + n2 = 0 + 166.6666667 Combine like terms: -166.6666667 + 166.6666667 = 0.0000000 0.0000000 + 55n + n2 = 0 + 166.6666667 55n + n2 = 0 + 166.6666667 Combine like terms: 0 + 166.6666667 = 166.6666667 55n + n2 = 166.6666667 The n term is 55n. Take half its coefficient (27.5). Square it (756.25) and add it to both sides. Add '756.25' to each side of the equation. 55n + 756.25 + n2 = 166.6666667 + 756.25 Reorder the terms: 756.25 + 55n + n2 = 166.6666667 + 756.25 Combine like terms: 166.6666667 + 756.25 = 922.9166667 756.25 + 55n + n2 = 922.9166667 Factor a perfect square on the left side: (n + 27.5)(n + 27.5) = 922.9166667 Calculate the square root of the right side: 30.379543556 Break this problem into two subproblems by setting (n + 27.5) equal to 30.379543556 and -30.379543556.Subproblem 1
n + 27.5 = 30.379543556 Simplifying n + 27.5 = 30.379543556 Reorder the terms: 27.5 + n = 30.379543556 Solving 27.5 + n = 30.379543556 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + n = 30.379543556 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + n = 30.379543556 + -27.5 n = 30.379543556 + -27.5 Combine like terms: 30.379543556 + -27.5 = 2.879543556 n = 2.879543556 Simplifying n = 2.879543556Subproblem 2
n + 27.5 = -30.379543556 Simplifying n + 27.5 = -30.379543556 Reorder the terms: 27.5 + n = -30.379543556 Solving 27.5 + n = -30.379543556 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + n = -30.379543556 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + n = -30.379543556 + -27.5 n = -30.379543556 + -27.5 Combine like terms: -30.379543556 + -27.5 = -57.879543556 n = -57.879543556 Simplifying n = -57.879543556Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.879543556, -57.879543556}
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