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Simplifying (n + -5)(n + -6) = 120 Reorder the terms: (-5 + n)(n + -6) = 120 Reorder the terms: (-5 + n)(-6 + n) = 120 Multiply (-5 + n) * (-6 + n) (-5(-6 + n) + n(-6 + n)) = 120 ((-6 * -5 + n * -5) + n(-6 + n)) = 120 ((30 + -5n) + n(-6 + n)) = 120 (30 + -5n + (-6 * n + n * n)) = 120 (30 + -5n + (-6n + n2)) = 120 Combine like terms: -5n + -6n = -11n (30 + -11n + n2) = 120 Solving 30 + -11n + n2 = 120 Solving for variable 'n'. Reorder the terms: 30 + -120 + -11n + n2 = 120 + -120 Combine like terms: 30 + -120 = -90 -90 + -11n + n2 = 120 + -120 Combine like terms: 120 + -120 = 0 -90 + -11n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '90' to each side of the equation. -90 + -11n + 90 + n2 = 0 + 90 Reorder the terms: -90 + 90 + -11n + n2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + -11n + n2 = 0 + 90 -11n + n2 = 0 + 90 Combine like terms: 0 + 90 = 90 -11n + n2 = 90 The n term is -11n. Take half its coefficient (-5.5). Square it (30.25) and add it to both sides. Add '30.25' to each side of the equation. -11n + 30.25 + n2 = 90 + 30.25 Reorder the terms: 30.25 + -11n + n2 = 90 + 30.25 Combine like terms: 90 + 30.25 = 120.25 30.25 + -11n + n2 = 120.25 Factor a perfect square on the left side: (n + -5.5)(n + -5.5) = 120.25 Calculate the square root of the right side: 10.9658561 Break this problem into two subproblems by setting (n + -5.5) equal to 10.9658561 and -10.9658561.Subproblem 1
n + -5.5 = 10.9658561 Simplifying n + -5.5 = 10.9658561 Reorder the terms: -5.5 + n = 10.9658561 Solving -5.5 + n = 10.9658561 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + n = 10.9658561 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = 10.9658561 + 5.5 n = 10.9658561 + 5.5 Combine like terms: 10.9658561 + 5.5 = 16.4658561 n = 16.4658561 Simplifying n = 16.4658561Subproblem 2
n + -5.5 = -10.9658561 Simplifying n + -5.5 = -10.9658561 Reorder the terms: -5.5 + n = -10.9658561 Solving -5.5 + n = -10.9658561 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5.5' to each side of the equation. -5.5 + 5.5 + n = -10.9658561 + 5.5 Combine like terms: -5.5 + 5.5 = 0.0 0.0 + n = -10.9658561 + 5.5 n = -10.9658561 + 5.5 Combine like terms: -10.9658561 + 5.5 = -5.4658561 n = -5.4658561 Simplifying n = -5.4658561Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.4658561, -5.4658561}
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