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Simplifying (n + 5)(n + 5 + -9) = 110 Reorder the terms: (5 + n)(n + 5 + -9) = 110 Reorder the terms: (5 + n)(5 + -9 + n) = 110 Combine like terms: 5 + -9 = -4 (5 + n)(-4 + n) = 110 Multiply (5 + n) * (-4 + n) (5(-4 + n) + n(-4 + n)) = 110 ((-4 * 5 + n * 5) + n(-4 + n)) = 110 ((-20 + 5n) + n(-4 + n)) = 110 (-20 + 5n + (-4 * n + n * n)) = 110 (-20 + 5n + (-4n + n2)) = 110 Combine like terms: 5n + -4n = 1n (-20 + 1n + n2) = 110 Solving -20 + 1n + n2 = 110 Solving for variable 'n'. Reorder the terms: -20 + -110 + 1n + n2 = 110 + -110 Combine like terms: -20 + -110 = -130 -130 + 1n + n2 = 110 + -110 Combine like terms: 110 + -110 = 0 -130 + 1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '130' to each side of the equation. -130 + 1n + 130 + n2 = 0 + 130 Reorder the terms: -130 + 130 + 1n + n2 = 0 + 130 Combine like terms: -130 + 130 = 0 0 + 1n + n2 = 0 + 130 1n + n2 = 0 + 130 Combine like terms: 0 + 130 = 130 1n + n2 = 130 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 = 130 + 0.25 Reorder the terms: 0.25 + n + n2 = 130 + 0.25 Combine like terms: 130 + 0.25 = 130.25 0.25 + n + n2 = 130.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 130.25 Calculate the square root of the right side: 11.412712211 Break this problem into two subproblems by setting (n + 0.5) equal to 11.412712211 and -11.412712211.Subproblem 1
n + 0.5 = 11.412712211 Simplifying n + 0.5 = 11.412712211 Reorder the terms: 0.5 + n = 11.412712211 Solving 0.5 + n = 11.412712211 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 = 11.412712211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 11.412712211 + -0.5 n = 11.412712211 + -0.5 Combine like terms: 11.412712211 + -0.5 = 10.912712211 n = 10.912712211 Simplifying n = 10.912712211Subproblem 2
n + 0.5 = -11.412712211 Simplifying n + 0.5 = -11.412712211 Reorder the terms: 0.5 + n = -11.412712211 Solving 0.5 + n = -11.412712211 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 = -11.412712211 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -11.412712211 + -0.5 n = -11.412712211 + -0.5 Combine like terms: -11.412712211 + -0.5 = -11.912712211 n = -11.912712211 Simplifying n = -11.912712211Solution
The solution to the problem is based on the solutions from the subproblems. n = {10.912712211, -11.912712211}
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