0=4n^2-16n-32

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Solution for 0=4n^2-16n-32 equation: 


Simplifying
0 = 4n2 + -16n + -32

Reorder the terms:
0 = -32 + -16n + 4n2

Solving
0 = -32 + -16n + 4n2

Solving for variable 'n'.

Combine like terms: 0 + 32 = 32
32 + 16n + -4n2 = -32 + -16n + 4n2 + 32 + 16n + -4n2

Reorder the terms:
32 + 16n + -4n2 = -32 + 32 + -16n + 16n + 4n2 + -4n2

Combine like terms: -32 + 32 = 0
32 + 16n + -4n2 = 0 + -16n + 16n + 4n2 + -4n2
32 + 16n + -4n2 = -16n + 16n + 4n2 + -4n2

Combine like terms: -16n + 16n = 0
32 + 16n + -4n2 = 0 + 4n2 + -4n2
32 + 16n + -4n2 = 4n2 + -4n2

Combine like terms: 4n2 + -4n2 = 0
32 + 16n + -4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(8 + 4n + -1n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(8 + 4n + -1n2)' equal to zero and attempt to solve:

Simplifying
8 + 4n + -1n2 = 0

Solving
8 + 4n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-8 + -4n + n2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -4n + 8 + n2 = 0 + 8

Reorder the terms:
-8 + 8 + -4n + n2 = 0 + 8

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

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

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 = 8 + 4

Reorder the terms:
4 + -4n + n2 = 8 + 4

Combine like terms: 8 + 4 = 12
4 + -4n + n2 = 12

Factor a perfect square on the left side:
(n + -2)(n + -2) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(n + -2) equal to 3.464101615 and -3.464101615.


Subproblem 1
n + -2 = 3.464101615

Simplifying
n + -2 = 3.464101615

Reorder the terms:
-2 + n = 3.464101615

Solving
-2 + n = 3.464101615

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 = 3.464101615 + 2

Combine like terms: -2 + 2 = 0
0 + n = 3.464101615 + 2
n = 3.464101615 + 2

Combine like terms: 3.464101615 + 2 = 5.464101615
n = 5.464101615

Simplifying
n = 5.464101615


Subproblem 2
n + -2 = -3.464101615

Simplifying
n + -2 = -3.464101615

Reorder the terms:
-2 + n = -3.464101615

Solving
-2 + n = -3.464101615

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 = -3.464101615 + 2

Combine like terms: -2 + 2 = 0
0 + n = -3.464101615 + 2
n = -3.464101615 + 2

Combine like terms: -3.464101615 + 2 = -1.464101615
n = -1.464101615

Simplifying
n = -1.464101615


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {5.464101615, -1.464101615}
Solution
n = {5.464101615, -1.464101615}

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