L^2+9L-180=0

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Solution for L^2+9L-180=0 equation: 


Simplifying
L2 + 9L + -180 = 0

Reorder the terms:
-180 + 9L + L2 = 0

Solving
-180 + 9L + L2 = 0

Solving for variable 'L'.

Begin completing the square.

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + 9L + 180 + L2 = 0 + 180

Reorder the terms:
-180 + 180 + 9L + L2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + 9L + L2 = 0 + 180
9L + L2 = 0 + 180

Combine like terms: 0 + 180 = 180
9L + L2 = 180

The L term is 9L.  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.
9L + 20.25 + L2 = 180 + 20.25

Reorder the terms:
20.25 + 9L + L2 = 180 + 20.25

Combine like terms: 180 + 20.25 = 200.25
20.25 + 9L + L2 = 200.25

Factor a perfect square on the left side:
(L + 4.5)(L + 4.5) = 200.25

Calculate the square root of the right side: 14.150971698

Break this problem into two subproblems by setting 
(L + 4.5) equal to 14.150971698 and -14.150971698.


Subproblem 1
L + 4.5 = 14.150971698

Simplifying
L + 4.5 = 14.150971698

Reorder the terms:
4.5 + L = 14.150971698

Solving
4.5 + L = 14.150971698

Solving for variable 'L'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + L = 14.150971698 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + L = 14.150971698 + -4.5
L = 14.150971698 + -4.5

Combine like terms: 14.150971698 + -4.5 = 9.650971698
L = 9.650971698

Simplifying
L = 9.650971698


Subproblem 2
L + 4.5 = -14.150971698

Simplifying
L + 4.5 = -14.150971698

Reorder the terms:
4.5 + L = -14.150971698

Solving
4.5 + L = -14.150971698

Solving for variable 'L'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + L = -14.150971698 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + L = -14.150971698 + -4.5
L = -14.150971698 + -4.5

Combine like terms: -14.150971698 + -4.5 = -18.650971698
L = -18.650971698

Simplifying
L = -18.650971698


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {9.650971698, -18.650971698}

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