2n(n+18)=24

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Solution for 2n(n+18)=24 equation: 


Simplifying
2n(n + 18) = 24

Reorder the terms:
2n(18 + n) = 24
(18 * 2n + n * 2n) = 24
(36n + 2n2) = 24

Solving
36n + 2n2 = 24

Solving for variable 'n'.

Reorder the terms:
-24 + 36n + 2n2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 36n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-12 + 18n + n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-12 + 18n + n2)' equal to zero and attempt to solve:

Simplifying
-12 + 18n + n2 = 0

Solving
-12 + 18n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 18n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 18n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 18n + n2 = 0 + 12
18n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
18n + n2 = 12

The n term is 18n.  Take half its coefficient (9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
18n + 81 + n2 = 12 + 81

Reorder the terms:
81 + 18n + n2 = 12 + 81

Combine like terms: 12 + 81 = 93
81 + 18n + n2 = 93

Factor a perfect square on the left side:
(n + 9)(n + 9) = 93

Calculate the square root of the right side: 9.643650761

Break this problem into two subproblems by setting 
(n + 9) equal to 9.643650761 and -9.643650761.


Subproblem 1
n + 9 = 9.643650761

Simplifying
n + 9 = 9.643650761

Reorder the terms:
9 + n = 9.643650761

Solving
9 + n = 9.643650761

Solving for variable 'n'.

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

Add '-9' to each side of the equation.
9 + -9 + n = 9.643650761 + -9

Combine like terms: 9 + -9 = 0
0 + n = 9.643650761 + -9
n = 9.643650761 + -9

Combine like terms: 9.643650761 + -9 = 0.643650761
n = 0.643650761

Simplifying
n = 0.643650761


Subproblem 2
n + 9 = -9.643650761

Simplifying
n + 9 = -9.643650761

Reorder the terms:
9 + n = -9.643650761

Solving
9 + n = -9.643650761

Solving for variable 'n'.

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

Add '-9' to each side of the equation.
9 + -9 + n = -9.643650761 + -9

Combine like terms: 9 + -9 = 0
0 + n = -9.643650761 + -9
n = -9.643650761 + -9

Combine like terms: -9.643650761 + -9 = -18.643650761
n = -18.643650761

Simplifying
n = -18.643650761


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.643650761, -18.643650761}
Solution
n = {0.643650761, -18.643650761}

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