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Simplifying (5n2 + -3 + n) + (2n2 + 5) + (2n2 + -5) = 0 Reorder the terms: (-3 + n + 5n2) + (2n2 + 5) + (2n2 + -5) = 0 Remove parenthesis around (-3 + n + 5n2) -3 + n + 5n2 + (2n2 + 5) + (2n2 + -5) = 0 Reorder the terms: -3 + n + 5n2 + (5 + 2n2) + (2n2 + -5) = 0 Remove parenthesis around (5 + 2n2) -3 + n + 5n2 + 5 + 2n2 + (2n2 + -5) = 0 Reorder the terms: -3 + n + 5n2 + 5 + 2n2 + (-5 + 2n2) = 0 Remove parenthesis around (-5 + 2n2) -3 + n + 5n2 + 5 + 2n2 + -5 + 2n2 = 0 Reorder the terms: -3 + 5 + -5 + n + 5n2 + 2n2 + 2n2 = 0 Combine like terms: -3 + 5 = 2 2 + -5 + n + 5n2 + 2n2 + 2n2 = 0 Combine like terms: 2 + -5 = -3 -3 + n + 5n2 + 2n2 + 2n2 = 0 Combine like terms: 5n2 + 2n2 = 7n2 -3 + n + 7n2 + 2n2 = 0 Combine like terms: 7n2 + 2n2 = 9n2 -3 + n + 9n2 = 0 Solving -3 + n + 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'. -0.3333333333 + 0.1111111111n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.1111111111n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.1111111111n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.1111111111n + n2 = 0 + 0.3333333333 0.1111111111n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.1111111111n + n2 = 0.3333333333 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1111111111n + 0.25 + n2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + 0.1111111111n + n2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + 0.1111111111n + n2 = 0.5833333333 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (n + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
n + 0.5 = 0.763762616 Simplifying n + 0.5 = 0.763762616 Reorder the terms: 0.5 + n = 0.763762616 Solving 0.5 + n = 0.763762616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.763762616 + -0.5 n = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 n = 0.263762616 Simplifying n = 0.263762616Subproblem 2
n + 0.5 = -0.763762616 Simplifying n + 0.5 = -0.763762616 Reorder the terms: 0.5 + n = -0.763762616 Solving 0.5 + n = -0.763762616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.763762616 + -0.5 n = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 n = -1.263762616 Simplifying n = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.263762616, -1.263762616}
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