75L^2-8040L-2224800=0

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Solution for 75L^2-8040L-2224800=0 equation: 


Simplifying
75L2 + -8040L + -2224800 = 0

Reorder the terms:
-2224800 + -8040L + 75L2 = 0

Solving
-2224800 + -8040L + 75L2 = 0

Solving for variable 'L'.

Factor out the Greatest Common Factor (GCF), '15'.
15(-148320 + -536L + 5L2) = 0

Ignore the factor 15.

Subproblem 1
Set the factor '(-148320 + -536L + 5L2)' equal to zero and attempt to solve:

Simplifying
-148320 + -536L + 5L2 = 0

Solving
-148320 + -536L + 5L2 = 0

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

Divide each side by '5'.
-29664 + -107.2L + L2 = 0

Move the constant term to the right:

Add '29664' to each side of the equation.
-29664 + -107.2L + 29664 + L2 = 0 + 29664

Reorder the terms:
-29664 + 29664 + -107.2L + L2 = 0 + 29664

Combine like terms: -29664 + 29664 = 0
0 + -107.2L + L2 = 0 + 29664
-107.2L + L2 = 0 + 29664

Combine like terms: 0 + 29664 = 29664
-107.2L + L2 = 29664

The L term is -107.2L.  Take half its coefficient (-53.6).
Square it (2872.96) and add it to both sides.

Add '2872.96' to each side of the equation.
-107.2L + 2872.96 + L2 = 29664 + 2872.96

Reorder the terms:
2872.96 + -107.2L + L2 = 29664 + 2872.96

Combine like terms: 29664 + 2872.96 = 32536.96
2872.96 + -107.2L + L2 = 32536.96

Factor a perfect square on the left side:
(L + -53.6)(L + -53.6) = 32536.96

Calculate the square root of the right side: 180.380043242

Break this problem into two subproblems by setting 
(L + -53.6) equal to 180.380043242 and -180.380043242.


Subproblem 1
L + -53.6 = 180.380043242

Simplifying
L + -53.6 = 180.380043242

Reorder the terms:
-53.6 + L = 180.380043242

Solving
-53.6 + L = 180.380043242

Solving for variable 'L'.

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

Add '53.6' to each side of the equation.
-53.6 + 53.6 + L = 180.380043242 + 53.6

Combine like terms: -53.6 + 53.6 = 0.0
0.0 + L = 180.380043242 + 53.6
L = 180.380043242 + 53.6

Combine like terms: 180.380043242 + 53.6 = 233.980043242
L = 233.980043242

Simplifying
L = 233.980043242


Subproblem 2
L + -53.6 = -180.380043242

Simplifying
L + -53.6 = -180.380043242

Reorder the terms:
-53.6 + L = -180.380043242

Solving
-53.6 + L = -180.380043242

Solving for variable 'L'.

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

Add '53.6' to each side of the equation.
-53.6 + 53.6 + L = -180.380043242 + 53.6

Combine like terms: -53.6 + 53.6 = 0.0
0.0 + L = -180.380043242 + 53.6
L = -180.380043242 + 53.6

Combine like terms: -180.380043242 + 53.6 = -126.780043242
L = -126.780043242

Simplifying
L = -126.780043242


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {233.980043242, -126.780043242}
Solution
L = {233.980043242, -126.780043242}

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