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Simplifying (n + 5)(n + 9) = 4 Reorder the terms: (5 + n)(n + 9) = 4 Reorder the terms: (5 + n)(9 + n) = 4 Multiply (5 + n) * (9 + n) (5(9 + n) + n(9 + n)) = 4 ((9 * 5 + n * 5) + n(9 + n)) = 4 ((45 + 5n) + n(9 + n)) = 4 (45 + 5n + (9 * n + n * n)) = 4 (45 + 5n + (9n + n2)) = 4 Combine like terms: 5n + 9n = 14n (45 + 14n + n2) = 4 Solving 45 + 14n + n2 = 4 Solving for variable 'n'. Reorder the terms: 45 + -4 + 14n + n2 = 4 + -4 Combine like terms: 45 + -4 = 41 41 + 14n + n2 = 4 + -4 Combine like terms: 4 + -4 = 0 41 + 14n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-41' to each side of the equation. 41 + 14n + -41 + n2 = 0 + -41 Reorder the terms: 41 + -41 + 14n + n2 = 0 + -41 Combine like terms: 41 + -41 = 0 0 + 14n + n2 = 0 + -41 14n + n2 = 0 + -41 Combine like terms: 0 + -41 = -41 14n + n2 = -41 The n term is 14n. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14n + 49 + n2 = -41 + 49 Reorder the terms: 49 + 14n + n2 = -41 + 49 Combine like terms: -41 + 49 = 8 49 + 14n + n2 = 8 Factor a perfect square on the left side: (n + 7)(n + 7) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (n + 7) equal to 2.828427125 and -2.828427125.Subproblem 1
n + 7 = 2.828427125 Simplifying n + 7 = 2.828427125 Reorder the terms: 7 + n = 2.828427125 Solving 7 + n = 2.828427125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = 2.828427125 + -7 Combine like terms: 7 + -7 = 0 0 + n = 2.828427125 + -7 n = 2.828427125 + -7 Combine like terms: 2.828427125 + -7 = -4.171572875 n = -4.171572875 Simplifying n = -4.171572875Subproblem 2
n + 7 = -2.828427125 Simplifying n + 7 = -2.828427125 Reorder the terms: 7 + n = -2.828427125 Solving 7 + n = -2.828427125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + n = -2.828427125 + -7 Combine like terms: 7 + -7 = 0 0 + n = -2.828427125 + -7 n = -2.828427125 + -7 Combine like terms: -2.828427125 + -7 = -9.828427125 n = -9.828427125 Simplifying n = -9.828427125Solution
The solution to the problem is based on the solutions from the subproblems. n = {-4.171572875, -9.828427125}
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