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Simplifying (n + 2)(n + 5) = 280 Reorder the terms: (2 + n)(n + 5) = 280 Reorder the terms: (2 + n)(5 + n) = 280 Multiply (2 + n) * (5 + n) (2(5 + n) + n(5 + n)) = 280 ((5 * 2 + n * 2) + n(5 + n)) = 280 ((10 + 2n) + n(5 + n)) = 280 (10 + 2n + (5 * n + n * n)) = 280 (10 + 2n + (5n + n2)) = 280 Combine like terms: 2n + 5n = 7n (10 + 7n + n2) = 280 Solving 10 + 7n + n2 = 280 Solving for variable 'n'. Reorder the terms: 10 + -280 + 7n + n2 = 280 + -280 Combine like terms: 10 + -280 = -270 -270 + 7n + n2 = 280 + -280 Combine like terms: 280 + -280 = 0 -270 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '270' to each side of the equation. -270 + 7n + 270 + n2 = 0 + 270 Reorder the terms: -270 + 270 + 7n + n2 = 0 + 270 Combine like terms: -270 + 270 = 0 0 + 7n + n2 = 0 + 270 7n + n2 = 0 + 270 Combine like terms: 0 + 270 = 270 7n + n2 = 270 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 270 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 270 + 12.25 Combine like terms: 270 + 12.25 = 282.25 12.25 + 7n + n2 = 282.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 282.25 Calculate the square root of the right side: 16.800297616 Break this problem into two subproblems by setting (n + 3.5) equal to 16.800297616 and -16.800297616.Subproblem 1
n + 3.5 = 16.800297616 Simplifying n + 3.5 = 16.800297616 Reorder the terms: 3.5 + n = 16.800297616 Solving 3.5 + n = 16.800297616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 16.800297616 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 16.800297616 + -3.5 n = 16.800297616 + -3.5 Combine like terms: 16.800297616 + -3.5 = 13.300297616 n = 13.300297616 Simplifying n = 13.300297616Subproblem 2
n + 3.5 = -16.800297616 Simplifying n + 3.5 = -16.800297616 Reorder the terms: 3.5 + n = -16.800297616 Solving 3.5 + n = -16.800297616 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -16.800297616 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -16.800297616 + -3.5 n = -16.800297616 + -3.5 Combine like terms: -16.800297616 + -3.5 = -20.300297616 n = -20.300297616 Simplifying n = -20.300297616Solution
The solution to the problem is based on the solutions from the subproblems. n = {13.300297616, -20.300297616}
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