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Simplifying (n + 6)(n + 6 + -7) = 49 Reorder the terms: (6 + n)(n + 6 + -7) = 49 Reorder the terms: (6 + n)(6 + -7 + n) = 49 Combine like terms: 6 + -7 = -1 (6 + n)(-1 + n) = 49 Multiply (6 + n) * (-1 + n) (6(-1 + n) + n(-1 + n)) = 49 ((-1 * 6 + n * 6) + n(-1 + n)) = 49 ((-6 + 6n) + n(-1 + n)) = 49 (-6 + 6n + (-1 * n + n * n)) = 49 (-6 + 6n + (-1n + n2)) = 49 Combine like terms: 6n + -1n = 5n (-6 + 5n + n2) = 49 Solving -6 + 5n + n2 = 49 Solving for variable 'n'. Reorder the terms: -6 + -49 + 5n + n2 = 49 + -49 Combine like terms: -6 + -49 = -55 -55 + 5n + n2 = 49 + -49 Combine like terms: 49 + -49 = 0 -55 + 5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '55' to each side of the equation. -55 + 5n + 55 + n2 = 0 + 55 Reorder the terms: -55 + 55 + 5n + n2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + 5n + n2 = 0 + 55 5n + n2 = 0 + 55 Combine like terms: 0 + 55 = 55 5n + n2 = 55 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 55 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 55 + 6.25 Combine like terms: 55 + 6.25 = 61.25 6.25 + 5n + n2 = 61.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 61.25 Calculate the square root of the right side: 7.826237921 Break this problem into two subproblems by setting (n + 2.5) equal to 7.826237921 and -7.826237921.Subproblem 1
n + 2.5 = 7.826237921 Simplifying n + 2.5 = 7.826237921 Reorder the terms: 2.5 + n = 7.826237921 Solving 2.5 + n = 7.826237921 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 7.826237921 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 7.826237921 + -2.5 n = 7.826237921 + -2.5 Combine like terms: 7.826237921 + -2.5 = 5.326237921 n = 5.326237921 Simplifying n = 5.326237921Subproblem 2
n + 2.5 = -7.826237921 Simplifying n + 2.5 = -7.826237921 Reorder the terms: 2.5 + n = -7.826237921 Solving 2.5 + n = -7.826237921 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -7.826237921 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -7.826237921 + -2.5 n = -7.826237921 + -2.5 Combine like terms: -7.826237921 + -2.5 = -10.326237921 n = -10.326237921 Simplifying n = -10.326237921Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.326237921, -10.326237921}
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