(3n+9)(2n+4)=84

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Solution for (3n+9)(2n+4)=84 equation:


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 = 1.274917218

Subproblem 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 = -6.274917218

Solution

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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