4n^2-72n-224=0

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Solution for 4n^2-72n-224=0 equation: 


Simplifying
4n2 + -72n + -224 = 0

Reorder the terms:
-224 + -72n + 4n2 = 0

Solving
-224 + -72n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-56 + -18n + n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-56 + -18n + n2)' equal to zero and attempt to solve:

Simplifying
-56 + -18n + n2 = 0

Solving
-56 + -18n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '56' to each side of the equation.
-56 + -18n + 56 + n2 = 0 + 56

Reorder the terms:
-56 + 56 + -18n + n2 = 0 + 56

Combine like terms: -56 + 56 = 0
0 + -18n + n2 = 0 + 56
-18n + n2 = 0 + 56

Combine like terms: 0 + 56 = 56
-18n + n2 = 56

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 = 56 + 81

Reorder the terms:
81 + -18n + n2 = 56 + 81

Combine like terms: 56 + 81 = 137
81 + -18n + n2 = 137

Factor a perfect square on the left side:
(n + -9)(n + -9) = 137

Calculate the square root of the right side: 11.704699911

Break this problem into two subproblems by setting 
(n + -9) equal to 11.704699911 and -11.704699911.


Subproblem 1
n + -9 = 11.704699911

Simplifying
n + -9 = 11.704699911

Reorder the terms:
-9 + n = 11.704699911

Solving
-9 + n = 11.704699911

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 = 11.704699911 + 9

Combine like terms: -9 + 9 = 0
0 + n = 11.704699911 + 9
n = 11.704699911 + 9

Combine like terms: 11.704699911 + 9 = 20.704699911
n = 20.704699911

Simplifying
n = 20.704699911


Subproblem 2
n + -9 = -11.704699911

Simplifying
n + -9 = -11.704699911

Reorder the terms:
-9 + n = -11.704699911

Solving
-9 + n = -11.704699911

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 = -11.704699911 + 9

Combine like terms: -9 + 9 = 0
0 + n = -11.704699911 + 9
n = -11.704699911 + 9

Combine like terms: -11.704699911 + 9 = -2.704699911
n = -2.704699911

Simplifying
n = -2.704699911


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {20.704699911, -2.704699911}
Solution
n = {20.704699911, -2.704699911}

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