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Simplifying 28n2 = -19 + -184n Solving 28n2 = -19 + -184n Solving for variable 'n'. Reorder the terms: 19 + 184n + 28n2 = -19 + -184n + 19 + 184n Reorder the terms: 19 + 184n + 28n2 = -19 + 19 + -184n + 184n Combine like terms: -19 + 19 = 0 19 + 184n + 28n2 = 0 + -184n + 184n 19 + 184n + 28n2 = -184n + 184n Combine like terms: -184n + 184n = 0 19 + 184n + 28n2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. 0.6785714286 + 6.571428571n + n2 = 0 Move the constant term to the right: Add '-0.6785714286' to each side of the equation. 0.6785714286 + 6.571428571n + -0.6785714286 + n2 = 0 + -0.6785714286 Reorder the terms: 0.6785714286 + -0.6785714286 + 6.571428571n + n2 = 0 + -0.6785714286 Combine like terms: 0.6785714286 + -0.6785714286 = 0.0000000000 0.0000000000 + 6.571428571n + n2 = 0 + -0.6785714286 6.571428571n + n2 = 0 + -0.6785714286 Combine like terms: 0 + -0.6785714286 = -0.6785714286 6.571428571n + n2 = -0.6785714286 The n term is 6.571428571n. Take half its coefficient (3.285714286). Square it (10.79591837) and add it to both sides. Add '10.79591837' to each side of the equation. 6.571428571n + 10.79591837 + n2 = -0.6785714286 + 10.79591837 Reorder the terms: 10.79591837 + 6.571428571n + n2 = -0.6785714286 + 10.79591837 Combine like terms: -0.6785714286 + 10.79591837 = 10.1173469414 10.79591837 + 6.571428571n + n2 = 10.1173469414 Factor a perfect square on the left side: (n + 3.285714286)(n + 3.285714286) = 10.1173469414 Calculate the square root of the right side: 3.180777726 Break this problem into two subproblems by setting (n + 3.285714286) equal to 3.180777726 and -3.180777726.Subproblem 1
n + 3.285714286 = 3.180777726 Simplifying n + 3.285714286 = 3.180777726 Reorder the terms: 3.285714286 + n = 3.180777726 Solving 3.285714286 + n = 3.180777726 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.285714286' to each side of the equation. 3.285714286 + -3.285714286 + n = 3.180777726 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = 3.180777726 + -3.285714286 n = 3.180777726 + -3.285714286 Combine like terms: 3.180777726 + -3.285714286 = -0.10493656 n = -0.10493656 Simplifying n = -0.10493656Subproblem 2
n + 3.285714286 = -3.180777726 Simplifying n + 3.285714286 = -3.180777726 Reorder the terms: 3.285714286 + n = -3.180777726 Solving 3.285714286 + n = -3.180777726 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.285714286' to each side of the equation. 3.285714286 + -3.285714286 + n = -3.180777726 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = -3.180777726 + -3.285714286 n = -3.180777726 + -3.285714286 Combine like terms: -3.180777726 + -3.285714286 = -6.466492012 n = -6.466492012 Simplifying n = -6.466492012Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.10493656, -6.466492012}
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