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Simplifying n2 + 2n + -48.3025 = 0 Reorder the terms: -48.3025 + 2n + n2 = 0 Solving -48.3025 + 2n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '48.3025' to each side of the equation. -48.3025 + 2n + 48.3025 + n2 = 0 + 48.3025 Reorder the terms: -48.3025 + 48.3025 + 2n + n2 = 0 + 48.3025 Combine like terms: -48.3025 + 48.3025 = 0.0000 0.0000 + 2n + n2 = 0 + 48.3025 2n + n2 = 0 + 48.3025 Combine like terms: 0 + 48.3025 = 48.3025 2n + n2 = 48.3025 The n term is 2n. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2n + 1 + n2 = 48.3025 + 1 Reorder the terms: 1 + 2n + n2 = 48.3025 + 1 Combine like terms: 48.3025 + 1 = 49.3025 1 + 2n + n2 = 49.3025 Factor a perfect square on the left side: (n + 1)(n + 1) = 49.3025 Calculate the square root of the right side: 7.021573898 Break this problem into two subproblems by setting (n + 1) equal to 7.021573898 and -7.021573898.Subproblem 1
n + 1 = 7.021573898 Simplifying n + 1 = 7.021573898 Reorder the terms: 1 + n = 7.021573898 Solving 1 + n = 7.021573898 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 7.021573898 + -1 Combine like terms: 1 + -1 = 0 0 + n = 7.021573898 + -1 n = 7.021573898 + -1 Combine like terms: 7.021573898 + -1 = 6.021573898 n = 6.021573898 Simplifying n = 6.021573898Subproblem 2
n + 1 = -7.021573898 Simplifying n + 1 = -7.021573898 Reorder the terms: 1 + n = -7.021573898 Solving 1 + n = -7.021573898 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -7.021573898 + -1 Combine like terms: 1 + -1 = 0 0 + n = -7.021573898 + -1 n = -7.021573898 + -1 Combine like terms: -7.021573898 + -1 = -8.021573898 n = -8.021573898 Simplifying n = -8.021573898Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.021573898, -8.021573898}
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