5+4n(n+9)=-3

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Solution for 5+4n(n+9)=-3 equation:


Simplifying
5 + 4n(n + 9) = -3

Reorder the terms:
5 + 4n(9 + n) = -3
5 + (9 * 4n + n * 4n) = -3
5 + (36n + 4n2) = -3

Solving
5 + 36n + 4n2 = -3

Solving for variable 'n'.

Reorder the terms:
5 + 3 + 36n + 4n2 = -3 + 3

Combine like terms: 5 + 3 = 8
8 + 36n + 4n2 = -3 + 3

Combine like terms: -3 + 3 = 0
8 + 36n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(2 + 9n + n2) = 0

Ignore the factor 4.

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

Simplifying
2 + 9n + n2 = 0

Solving
2 + 9n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 9n + -2 + n2 = 0 + -2

Reorder the terms:
2 + -2 + 9n + n2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 9n + n2 = 0 + -2
9n + n2 = 0 + -2

Combine like terms: 0 + -2 = -2
9n + n2 = -2

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 = -2 + 20.25

Reorder the terms:
20.25 + 9n + n2 = -2 + 20.25

Combine like terms: -2 + 20.25 = 18.25
20.25 + 9n + n2 = 18.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + 4.5) equal to 4.272001873 and -4.272001873.


Subproblem 1
n + 4.5 = 4.272001873

Simplifying
n + 4.5 = 4.272001873

Reorder the terms:
4.5 + n = 4.272001873

Solving
4.5 + n = 4.272001873

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 = 4.272001873 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + n = 4.272001873 + -4.5
n = 4.272001873 + -4.5

Combine like terms: 4.272001873 + -4.5 = -0.227998127
n = -0.227998127

Simplifying
n = -0.227998127


Subproblem 2
n + 4.5 = -4.272001873

Simplifying
n + 4.5 = -4.272001873

Reorder the terms:
4.5 + n = -4.272001873

Solving
4.5 + n = -4.272001873

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 = -4.272001873 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + n = -4.272001873 + -4.5
n = -4.272001873 + -4.5

Combine like terms: -4.272001873 + -4.5 = -8.772001873
n = -8.772001873

Simplifying
n = -8.772001873


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.227998127, -8.772001873}
Solution
n = {-0.227998127, -8.772001873}

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