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Simplifying (n + 7)(n + 9) = 1000 Reorder the terms: (7 + n)(n + 9) = 1000 Reorder the terms: (7 + n)(9 + n) = 1000 Multiply (7 + n) * (9 + n) (7(9 + n) + n(9 + n)) = 1000 ((9 * 7 + n * 7) + n(9 + n)) = 1000 ((63 + 7n) + n(9 + n)) = 1000 (63 + 7n + (9 * n + n * n)) = 1000 (63 + 7n + (9n + n2)) = 1000 Combine like terms: 7n + 9n = 16n (63 + 16n + n2) = 1000 Solving 63 + 16n + n2 = 1000 Solving for variable 'n'. Reorder the terms: 63 + -1000 + 16n + n2 = 1000 + -1000 Combine like terms: 63 + -1000 = -937 -937 + 16n + n2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -937 + 16n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '937' to each side of the equation. -937 + 16n + 937 + n2 = 0 + 937 Reorder the terms: -937 + 937 + 16n + n2 = 0 + 937 Combine like terms: -937 + 937 = 0 0 + 16n + n2 = 0 + 937 16n + n2 = 0 + 937 Combine like terms: 0 + 937 = 937 16n + n2 = 937 The n term is 16n. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16n + 64 + n2 = 937 + 64 Reorder the terms: 64 + 16n + n2 = 937 + 64 Combine like terms: 937 + 64 = 1001 64 + 16n + n2 = 1001 Factor a perfect square on the left side: (n + 8)(n + 8) = 1001 Calculate the square root of the right side: 31.638584039 Break this problem into two subproblems by setting (n + 8) equal to 31.638584039 and -31.638584039.Subproblem 1
n + 8 = 31.638584039 Simplifying n + 8 = 31.638584039 Reorder the terms: 8 + n = 31.638584039 Solving 8 + n = 31.638584039 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = 31.638584039 + -8 Combine like terms: 8 + -8 = 0 0 + n = 31.638584039 + -8 n = 31.638584039 + -8 Combine like terms: 31.638584039 + -8 = 23.638584039 n = 23.638584039 Simplifying n = 23.638584039Subproblem 2
n + 8 = -31.638584039 Simplifying n + 8 = -31.638584039 Reorder the terms: 8 + n = -31.638584039 Solving 8 + n = -31.638584039 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + n = -31.638584039 + -8 Combine like terms: 8 + -8 = 0 0 + n = -31.638584039 + -8 n = -31.638584039 + -8 Combine like terms: -31.638584039 + -8 = -39.638584039 n = -39.638584039 Simplifying n = -39.638584039Solution
The solution to the problem is based on the solutions from the subproblems. n = {23.638584039, -39.638584039}
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