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Simplifying 2n2 + n + -59 = -3 Reorder the terms: -59 + n + 2n2 = -3 Solving -59 + n + 2n2 = -3 Solving for variable 'n'. Reorder the terms: -59 + 3 + n + 2n2 = -3 + 3 Combine like terms: -59 + 3 = -56 -56 + n + 2n2 = -3 + 3 Combine like terms: -3 + 3 = 0 -56 + n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -28 + 0.5n + n2 = 0 Move the constant term to the right: Add '28' to each side of the equation. -28 + 0.5n + 28 + n2 = 0 + 28 Reorder the terms: -28 + 28 + 0.5n + n2 = 0 + 28 Combine like terms: -28 + 28 = 0 0 + 0.5n + n2 = 0 + 28 0.5n + n2 = 0 + 28 Combine like terms: 0 + 28 = 28 0.5n + n2 = 28 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. 0.5n + 0.25 + n2 = 28 + 0.25 Reorder the terms: 0.25 + 0.5n + n2 = 28 + 0.25 Combine like terms: 28 + 0.25 = 28.25 0.25 + 0.5n + n2 = 28.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 28.25 Calculate the square root of the right side: 5.315072906 Break this problem into two subproblems by setting (n + 0.5) equal to 5.315072906 and -5.315072906.Subproblem 1
n + 0.5 = 5.315072906 Simplifying n + 0.5 = 5.315072906 Reorder the terms: 0.5 + n = 5.315072906 Solving 0.5 + n = 5.315072906 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 = 5.315072906 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 5.315072906 + -0.5 n = 5.315072906 + -0.5 Combine like terms: 5.315072906 + -0.5 = 4.815072906 n = 4.815072906 Simplifying n = 4.815072906Subproblem 2
n + 0.5 = -5.315072906 Simplifying n + 0.5 = -5.315072906 Reorder the terms: 0.5 + n = -5.315072906 Solving 0.5 + n = -5.315072906 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 = -5.315072906 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -5.315072906 + -0.5 n = -5.315072906 + -0.5 Combine like terms: -5.315072906 + -0.5 = -5.815072906 n = -5.815072906 Simplifying n = -5.815072906Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.815072906, -5.815072906}
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