2n(n+9)=84

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

Simplifying
2n(n + 9) = 84

Reorder the terms:
2n(9 + n) = 84
(9 * 2n + n * 2n) = 84
(18n + 2n2) = 84

Solving
18n + 2n2 = 84

Solving for variable 'n'.

Reorder the terms:
-84 + 18n + 2n2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 18n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 9n + n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-42 + 9n + n2)' equal to zero and attempt to solve:

Simplifying
-42 + 9n + n2 = 0

Solving
-42 + 9n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + 9n + 42 + n2 = 0 + 42

Reorder the terms:
-42 + 42 + 9n + n2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + 9n + n2 = 0 + 42
9n + n2 = 0 + 42

Combine like terms: 0 + 42 = 42
9n + n2 = 42

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = 42 + 20.25

Reorder the terms:
20.25 + 9n + n2 = 42 + 20.25

Combine like terms: 42 + 20.25 = 62.25
20.25 + 9n + n2 = 62.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 62.25

Calculate the square root of the right side: 7.889866919

Break this problem into two subproblems by setting 
(n + 4.5) equal to 7.889866919 and -7.889866919.

Subproblem 1
n + 4.5 = 7.889866919

Simplifying
n + 4.5 = 7.889866919

Reorder the terms:
4.5 + n = 7.889866919

Solving
4.5 + n = 7.889866919

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + n = 7.889866919 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + n = 7.889866919 + -4.5
n = 7.889866919 + -4.5

Combine like terms: 7.889866919 + -4.5 = 3.389866919
n = 3.389866919

Simplifying
n = 3.389866919

Subproblem 2
n + 4.5 = -7.889866919

Simplifying
n + 4.5 = -7.889866919

Reorder the terms:
4.5 + n = -7.889866919

Solving
4.5 + n = -7.889866919

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + n = -7.889866919 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + n = -7.889866919 + -4.5
n = -7.889866919 + -4.5

Combine like terms: -7.889866919 + -4.5 = -12.389866919
n = -12.389866919

Simplifying
n = -12.389866919

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {3.389866919, -12.389866919}
Solution
n = {3.389866919, -12.389866919}