3-2n(n+4)=0

Solution for 3-2n(n+4)=0 equation:

Simplifying
3 + -2n(n + 4) = 0

Reorder the terms:
3 + -2n(4 + n) = 0
3 + (4 * -2n + n * -2n) = 0
3 + (-8n + -2n2) = 0

Solving
3 + -8n + -2n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1.5 + 4n + n2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 4n + 1.5 + n2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 4n + n2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 4n + n2 = 0 + 1.5
4n + n2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
4n + n2 = 1.5

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = 1.5 + 4

Reorder the terms:
4 + 4n + n2 = 1.5 + 4

Combine like terms: 1.5 + 4 = 5.5
4 + 4n + n2 = 5.5

Factor a perfect square on the left side:
(n + 2)(n + 2) = 5.5

Calculate the square root of the right side: 2.34520788

Break this problem into two subproblems by setting 
(n + 2) equal to 2.34520788 and -2.34520788.

Subproblem 1
n + 2 = 2.34520788

Simplifying
n + 2 = 2.34520788

Reorder the terms:
2 + n = 2.34520788

Solving
2 + n = 2.34520788

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = 2.34520788 + -2

Combine like terms: 2 + -2 = 0
0 + n = 2.34520788 + -2
n = 2.34520788 + -2

Combine like terms: 2.34520788 + -2 = 0.34520788
n = 0.34520788

Simplifying
n = 0.34520788

Subproblem 2
n + 2 = -2.34520788

Simplifying
n + 2 = -2.34520788

Reorder the terms:
2 + n = -2.34520788

Solving
2 + n = -2.34520788

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = -2.34520788 + -2

Combine like terms: 2 + -2 = 0
0 + n = -2.34520788 + -2
n = -2.34520788 + -2

Combine like terms: -2.34520788 + -2 = -4.34520788
n = -4.34520788

Simplifying
n = -4.34520788

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.34520788, -4.34520788}