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Simplifying (n + 7)(n + 2) + -1(n + 5) = 66 Reorder the terms: (7 + n)(n + 2) + -1(n + 5) = 66 Reorder the terms: (7 + n)(2 + n) + -1(n + 5) = 66 Multiply (7 + n) * (2 + n) (7(2 + n) + n(2 + n)) + -1(n + 5) = 66 ((2 * 7 + n * 7) + n(2 + n)) + -1(n + 5) = 66 ((14 + 7n) + n(2 + n)) + -1(n + 5) = 66 (14 + 7n + (2 * n + n * n)) + -1(n + 5) = 66 (14 + 7n + (2n + n2)) + -1(n + 5) = 66 Combine like terms: 7n + 2n = 9n (14 + 9n + n2) + -1(n + 5) = 66 Reorder the terms: 14 + 9n + n2 + -1(5 + n) = 66 14 + 9n + n2 + (5 * -1 + n * -1) = 66 14 + 9n + n2 + (-5 + -1n) = 66 Reorder the terms: 14 + -5 + 9n + -1n + n2 = 66 Combine like terms: 14 + -5 = 9 9 + 9n + -1n + n2 = 66 Combine like terms: 9n + -1n = 8n 9 + 8n + n2 = 66 Solving 9 + 8n + n2 = 66 Solving for variable 'n'. Reorder the terms: 9 + -66 + 8n + n2 = 66 + -66 Combine like terms: 9 + -66 = -57 -57 + 8n + n2 = 66 + -66 Combine like terms: 66 + -66 = 0 -57 + 8n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '57' to each side of the equation. -57 + 8n + 57 + n2 = 0 + 57 Reorder the terms: -57 + 57 + 8n + n2 = 0 + 57 Combine like terms: -57 + 57 = 0 0 + 8n + n2 = 0 + 57 8n + n2 = 0 + 57 Combine like terms: 0 + 57 = 57 8n + n2 = 57 The n term is 8n. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8n + 16 + n2 = 57 + 16 Reorder the terms: 16 + 8n + n2 = 57 + 16 Combine like terms: 57 + 16 = 73 16 + 8n + n2 = 73 Factor a perfect square on the left side: (n + 4)(n + 4) = 73 Calculate the square root of the right side: 8.544003745 Break this problem into two subproblems by setting (n + 4) equal to 8.544003745 and -8.544003745.Subproblem 1
n + 4 = 8.544003745 Simplifying n + 4 = 8.544003745 Reorder the terms: 4 + n = 8.544003745 Solving 4 + n = 8.544003745 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = 8.544003745 + -4 Combine like terms: 4 + -4 = 0 0 + n = 8.544003745 + -4 n = 8.544003745 + -4 Combine like terms: 8.544003745 + -4 = 4.544003745 n = 4.544003745 Simplifying n = 4.544003745Subproblem 2
n + 4 = -8.544003745 Simplifying n + 4 = -8.544003745 Reorder the terms: 4 + n = -8.544003745 Solving 4 + n = -8.544003745 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + n = -8.544003745 + -4 Combine like terms: 4 + -4 = 0 0 + n = -8.544003745 + -4 n = -8.544003745 + -4 Combine like terms: -8.544003745 + -4 = -12.544003745 n = -12.544003745 Simplifying n = -12.544003745Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.544003745, -12.544003745}
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