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Simplifying (n + -5)(n + 30) = 48 Reorder the terms: (-5 + n)(n + 30) = 48 Reorder the terms: (-5 + n)(30 + n) = 48 Multiply (-5 + n) * (30 + n) (-5(30 + n) + n(30 + n)) = 48 ((30 * -5 + n * -5) + n(30 + n)) = 48 ((-150 + -5n) + n(30 + n)) = 48 (-150 + -5n + (30 * n + n * n)) = 48 (-150 + -5n + (30n + n2)) = 48 Combine like terms: -5n + 30n = 25n (-150 + 25n + n2) = 48 Solving -150 + 25n + n2 = 48 Solving for variable 'n'. Reorder the terms: -150 + -48 + 25n + n2 = 48 + -48 Combine like terms: -150 + -48 = -198 -198 + 25n + n2 = 48 + -48 Combine like terms: 48 + -48 = 0 -198 + 25n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '198' to each side of the equation. -198 + 25n + 198 + n2 = 0 + 198 Reorder the terms: -198 + 198 + 25n + n2 = 0 + 198 Combine like terms: -198 + 198 = 0 0 + 25n + n2 = 0 + 198 25n + n2 = 0 + 198 Combine like terms: 0 + 198 = 198 25n + n2 = 198 The n term is 25n. Take half its coefficient (12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. 25n + 156.25 + n2 = 198 + 156.25 Reorder the terms: 156.25 + 25n + n2 = 198 + 156.25 Combine like terms: 198 + 156.25 = 354.25 156.25 + 25n + n2 = 354.25 Factor a perfect square on the left side: (n + 12.5)(n + 12.5) = 354.25 Calculate the square root of the right side: 18.821530225 Break this problem into two subproblems by setting (n + 12.5) equal to 18.821530225 and -18.821530225.Subproblem 1
n + 12.5 = 18.821530225 Simplifying n + 12.5 = 18.821530225 Reorder the terms: 12.5 + n = 18.821530225 Solving 12.5 + n = 18.821530225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + n = 18.821530225 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + n = 18.821530225 + -12.5 n = 18.821530225 + -12.5 Combine like terms: 18.821530225 + -12.5 = 6.321530225 n = 6.321530225 Simplifying n = 6.321530225Subproblem 2
n + 12.5 = -18.821530225 Simplifying n + 12.5 = -18.821530225 Reorder the terms: 12.5 + n = -18.821530225 Solving 12.5 + n = -18.821530225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + n = -18.821530225 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + n = -18.821530225 + -12.5 n = -18.821530225 + -12.5 Combine like terms: -18.821530225 + -12.5 = -31.321530225 n = -31.321530225 Simplifying n = -31.321530225Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.321530225, -31.321530225}
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