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Simplifying n(n + 2) + (n + 4) = 42 Reorder the terms: n(2 + n) + (n + 4) = 42 (2 * n + n * n) + (n + 4) = 42 (2n + n2) + (n + 4) = 42 Reorder the terms: 2n + n2 + (4 + n) = 42 Remove parenthesis around (4 + n) 2n + n2 + 4 + n = 42 Reorder the terms: 4 + 2n + n + n2 = 42 Combine like terms: 2n + n = 3n 4 + 3n + n2 = 42 Solving 4 + 3n + n2 = 42 Solving for variable 'n'. Reorder the terms: 4 + -42 + 3n + n2 = 42 + -42 Combine like terms: 4 + -42 = -38 -38 + 3n + n2 = 42 + -42 Combine like terms: 42 + -42 = 0 -38 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '38' to each side of the equation. -38 + 3n + 38 + n2 = 0 + 38 Reorder the terms: -38 + 38 + 3n + n2 = 0 + 38 Combine like terms: -38 + 38 = 0 0 + 3n + n2 = 0 + 38 3n + n2 = 0 + 38 Combine like terms: 0 + 38 = 38 3n + n2 = 38 The n term is 3n. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3n + 2.25 + n2 = 38 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 38 + 2.25 Combine like terms: 38 + 2.25 = 40.25 2.25 + 3n + n2 = 40.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 40.25 Calculate the square root of the right side: 6.34428877 Break this problem into two subproblems by setting (n + 1.5) equal to 6.34428877 and -6.34428877.Subproblem 1
n + 1.5 = 6.34428877 Simplifying n + 1.5 = 6.34428877 Reorder the terms: 1.5 + n = 6.34428877 Solving 1.5 + n = 6.34428877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 6.34428877 + -1.5 n = 6.34428877 + -1.5 Combine like terms: 6.34428877 + -1.5 = 4.84428877 n = 4.84428877 Simplifying n = 4.84428877Subproblem 2
n + 1.5 = -6.34428877 Simplifying n + 1.5 = -6.34428877 Reorder the terms: 1.5 + n = -6.34428877 Solving 1.5 + n = -6.34428877 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -6.34428877 + -1.5 n = -6.34428877 + -1.5 Combine like terms: -6.34428877 + -1.5 = -7.84428877 n = -7.84428877 Simplifying n = -7.84428877Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.84428877, -7.84428877}
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