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Simplifying (3n + 9)(2n + 4) = 84 Reorder the terms: (9 + 3n)(2n + 4) = 84 Reorder the terms: (9 + 3n)(4 + 2n) = 84 Multiply (9 + 3n) * (4 + 2n) (9(4 + 2n) + 3n * (4 + 2n)) = 84 ((4 * 9 + 2n * 9) + 3n * (4 + 2n)) = 84 ((36 + 18n) + 3n * (4 + 2n)) = 84 (36 + 18n + (4 * 3n + 2n * 3n)) = 84 (36 + 18n + (12n + 6n2)) = 84 Combine like terms: 18n + 12n = 30n (36 + 30n + 6n2) = 84 Solving 36 + 30n + 6n2 = 84 Solving for variable 'n'. Reorder the terms: 36 + -84 + 30n + 6n2 = 84 + -84 Combine like terms: 36 + -84 = -48 -48 + 30n + 6n2 = 84 + -84 Combine like terms: 84 + -84 = 0 -48 + 30n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-8 + 5n + n2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-8 + 5n + n2)' equal to zero and attempt to solve: Simplifying -8 + 5n + n2 = 0 Solving -8 + 5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '8' to each side of the equation. -8 + 5n + 8 + n2 = 0 + 8 Reorder the terms: -8 + 8 + 5n + n2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 5n + n2 = 0 + 8 5n + n2 = 0 + 8 Combine like terms: 0 + 8 = 8 5n + n2 = 8 The n term is 5n. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5n + 6.25 + n2 = 8 + 6.25 Reorder the terms: 6.25 + 5n + n2 = 8 + 6.25 Combine like terms: 8 + 6.25 = 14.25 6.25 + 5n + n2 = 14.25 Factor a perfect square on the left side: (n + 2.5)(n + 2.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (n + 2.5) equal to 3.774917218 and -3.774917218.Subproblem 1
n + 2.5 = 3.774917218 Simplifying n + 2.5 = 3.774917218 Reorder the terms: 2.5 + n = 3.774917218 Solving 2.5 + n = 3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = 3.774917218 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = 3.774917218 + -2.5 n = 3.774917218 + -2.5 Combine like terms: 3.774917218 + -2.5 = 1.274917218 n = 1.274917218 Simplifying n = 1.274917218Subproblem 2
n + 2.5 = -3.774917218 Simplifying n + 2.5 = -3.774917218 Reorder the terms: 2.5 + n = -3.774917218 Solving 2.5 + n = -3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + n = -3.774917218 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + n = -3.774917218 + -2.5 n = -3.774917218 + -2.5 Combine like terms: -3.774917218 + -2.5 = -6.274917218 n = -6.274917218 Simplifying n = -6.274917218Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.274917218, -6.274917218}Solution
n = {1.274917218, -6.274917218}
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