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Simplifying (n + 6)(n + -5) = 15 Reorder the terms: (6 + n)(n + -5) = 15 Reorder the terms: (6 + n)(-5 + n) = 15 Multiply (6 + n) * (-5 + n) (6(-5 + n) + n(-5 + n)) = 15 ((-5 * 6 + n * 6) + n(-5 + n)) = 15 ((-30 + 6n) + n(-5 + n)) = 15 (-30 + 6n + (-5 * n + n * n)) = 15 (-30 + 6n + (-5n + n2)) = 15 Combine like terms: 6n + -5n = 1n (-30 + 1n + n2) = 15 Solving -30 + 1n + n2 = 15 Solving for variable 'n'. Reorder the terms: -30 + -15 + 1n + n2 = 15 + -15 Combine like terms: -30 + -15 = -45 -45 + 1n + n2 = 15 + -15 Combine like terms: 15 + -15 = 0 -45 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '45' to each side of the equation. -45 + 1n + 45 + n2 = 0 + 45 Reorder the terms: -45 + 45 + 1n + n2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 1n + n2 = 0 + 45 1n + n2 = 0 + 45 Combine like terms: 0 + 45 = 45 1n + n2 = 45 The n term is 1n. 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. n + 0.25 + n2 = 45 + 0.25 Reorder the terms: 0.25 + n + n2 = 45 + 0.25 Combine like terms: 45 + 0.25 = 45.25 0.25 + n + n2 = 45.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 45.25 Calculate the square root of the right side: 6.726812024 Break this problem into two subproblems by setting (n + 0.5) equal to 6.726812024 and -6.726812024.Subproblem 1
n + 0.5 = 6.726812024 Simplifying n + 0.5 = 6.726812024 Reorder the terms: 0.5 + n = 6.726812024 Solving 0.5 + n = 6.726812024 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 = 6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 6.726812024 + -0.5 n = 6.726812024 + -0.5 Combine like terms: 6.726812024 + -0.5 = 6.226812024 n = 6.226812024 Simplifying n = 6.226812024Subproblem 2
n + 0.5 = -6.726812024 Simplifying n + 0.5 = -6.726812024 Reorder the terms: 0.5 + n = -6.726812024 Solving 0.5 + n = -6.726812024 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 = -6.726812024 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -6.726812024 + -0.5 n = -6.726812024 + -0.5 Combine like terms: -6.726812024 + -0.5 = -7.226812024 n = -7.226812024 Simplifying n = -7.226812024Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.226812024, -7.226812024}
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