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Simplifying 28n2 = -97 + -184n Solving 28n2 = -97 + -184n Solving for variable 'n'. Reorder the terms: 97 + 184n + 28n2 = -97 + -184n + 97 + 184n Reorder the terms: 97 + 184n + 28n2 = -97 + 97 + -184n + 184n Combine like terms: -97 + 97 = 0 97 + 184n + 28n2 = 0 + -184n + 184n 97 + 184n + 28n2 = -184n + 184n Combine like terms: -184n + 184n = 0 97 + 184n + 28n2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. 3.464285714 + 6.571428571n + n2 = 0 Move the constant term to the right: Add '-3.464285714' to each side of the equation. 3.464285714 + 6.571428571n + -3.464285714 + n2 = 0 + -3.464285714 Reorder the terms: 3.464285714 + -3.464285714 + 6.571428571n + n2 = 0 + -3.464285714 Combine like terms: 3.464285714 + -3.464285714 = 0.000000000 0.000000000 + 6.571428571n + n2 = 0 + -3.464285714 6.571428571n + n2 = 0 + -3.464285714 Combine like terms: 0 + -3.464285714 = -3.464285714 6.571428571n + n2 = -3.464285714 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 = -3.464285714 + 10.79591837 Reorder the terms: 10.79591837 + 6.571428571n + n2 = -3.464285714 + 10.79591837 Combine like terms: -3.464285714 + 10.79591837 = 7.331632656 10.79591837 + 6.571428571n + n2 = 7.331632656 Factor a perfect square on the left side: (n + 3.285714286)(n + 3.285714286) = 7.331632656 Calculate the square root of the right side: 2.707698775 Break this problem into two subproblems by setting (n + 3.285714286) equal to 2.707698775 and -2.707698775.Subproblem 1
n + 3.285714286 = 2.707698775 Simplifying n + 3.285714286 = 2.707698775 Reorder the terms: 3.285714286 + n = 2.707698775 Solving 3.285714286 + n = 2.707698775 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 = 2.707698775 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = 2.707698775 + -3.285714286 n = 2.707698775 + -3.285714286 Combine like terms: 2.707698775 + -3.285714286 = -0.578015511 n = -0.578015511 Simplifying n = -0.578015511Subproblem 2
n + 3.285714286 = -2.707698775 Simplifying n + 3.285714286 = -2.707698775 Reorder the terms: 3.285714286 + n = -2.707698775 Solving 3.285714286 + n = -2.707698775 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 = -2.707698775 + -3.285714286 Combine like terms: 3.285714286 + -3.285714286 = 0.000000000 0.000000000 + n = -2.707698775 + -3.285714286 n = -2.707698775 + -3.285714286 Combine like terms: -2.707698775 + -3.285714286 = -5.993413061 n = -5.993413061 Simplifying n = -5.993413061Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.578015511, -5.993413061}
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