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Simplifying (n + 2)(n + 4) = 42 Reorder the terms: (2 + n)(n + 4) = 42 Reorder the terms: (2 + n)(4 + n) = 42 Multiply (2 + n) * (4 + n) (2(4 + n) + n(4 + n)) = 42 ((4 * 2 + n * 2) + n(4 + n)) = 42 ((8 + 2n) + n(4 + n)) = 42 (8 + 2n + (4 * n + n * n)) = 42 (8 + 2n + (4n + n2)) = 42 Combine like terms: 2n + 4n = 6n (8 + 6n + n2) = 42 Solving 8 + 6n + n2 = 42 Solving for variable 'n'. Reorder the terms: 8 + -42 + 6n + n2 = 42 + -42 Combine like terms: 8 + -42 = -34 -34 + 6n + n2 = 42 + -42 Combine like terms: 42 + -42 = 0 -34 + 6n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '34' to each side of the equation. -34 + 6n + 34 + n2 = 0 + 34 Reorder the terms: -34 + 34 + 6n + n2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 6n + n2 = 0 + 34 6n + n2 = 0 + 34 Combine like terms: 0 + 34 = 34 6n + n2 = 34 The n term is 6n. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6n + 9 + n2 = 34 + 9 Reorder the terms: 9 + 6n + n2 = 34 + 9 Combine like terms: 34 + 9 = 43 9 + 6n + n2 = 43 Factor a perfect square on the left side: (n + 3)(n + 3) = 43 Calculate the square root of the right side: 6.557438524 Break this problem into two subproblems by setting (n + 3) equal to 6.557438524 and -6.557438524.Subproblem 1
n + 3 = 6.557438524 Simplifying n + 3 = 6.557438524 Reorder the terms: 3 + n = 6.557438524 Solving 3 + n = 6.557438524 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 6.557438524 + -3 Combine like terms: 3 + -3 = 0 0 + n = 6.557438524 + -3 n = 6.557438524 + -3 Combine like terms: 6.557438524 + -3 = 3.557438524 n = 3.557438524 Simplifying n = 3.557438524Subproblem 2
n + 3 = -6.557438524 Simplifying n + 3 = -6.557438524 Reorder the terms: 3 + n = -6.557438524 Solving 3 + n = -6.557438524 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -6.557438524 + -3 Combine like terms: 3 + -3 = 0 0 + n = -6.557438524 + -3 n = -6.557438524 + -3 Combine like terms: -6.557438524 + -3 = -9.557438524 n = -9.557438524 Simplifying n = -9.557438524Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.557438524, -9.557438524}
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