0=(-L^2)+11.5L-33

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Solution for 0=(-L^2)+11.5L-33 equation: 


Simplifying
0 = (-1L2) + 11.5L + -33

Reorder the terms:
0 = -33 + 11.5L + (-1L2)

Solving
0 = -33 + 11.5L + (-1L2)

Solving for variable 'L'.
0 + 33 + -11.5L + L2 = -33 + 11.5L + (-1L2) + 33 + -11.5L + (L2)

Combine like terms: 0 + 33 = 33
33 + -11.5L + L2 = -33 + 11.5L + (-1L2) + 33 + -11.5L + (L2)
33 + -11.5L + L2 = -33 + 11.5L + (-1L2) + 33 + -11.5L + L2

Reorder the terms:
33 + -11.5L + L2 = -33 + 33 + 11.5L + -11.5L + (-1L2) + L2

Combine like terms: -33 + 33 = 0
33 + -11.5L + L2 = 0 + 11.5L + -11.5L + (-1L2) + L2
33 + -11.5L + L2 = 11.5L + -11.5L + (-1L2) + L2

Combine like terms: 11.5L + -11.5L = 0.0
33 + -11.5L + L2 = 0.0 + (-1L2) + L2
33 + -11.5L + L2 = (-1L2) + L2

Combine like terms: (-1L2) + L2 = 0
33 + -11.5L + L2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-33' to each side of the equation.
33 + -11.5L + -33 + L2 = 0 + -33

Reorder the terms:
33 + -33 + -11.5L + L2 = 0 + -33

Combine like terms: 33 + -33 = 0
0 + -11.5L + L2 = 0 + -33
-11.5L + L2 = 0 + -33

Combine like terms: 0 + -33 = -33
-11.5L + L2 = -33

The L term is -11.5L.  Take half its coefficient (-5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
-11.5L + 33.0625 + L2 = -33 + 33.0625

Reorder the terms:
33.0625 + -11.5L + L2 = -33 + 33.0625

Combine like terms: -33 + 33.0625 = 0.0625
33.0625 + -11.5L + L2 = 0.0625

Factor a perfect square on the left side:
(L + -5.75)(L + -5.75) = 0.0625

Calculate the square root of the right side: 0.25

Break this problem into two subproblems by setting 
(L + -5.75) equal to 0.25 and -0.25.


Subproblem 1
L + -5.75 = 0.25

Simplifying
L + -5.75 = 0.25

Reorder the terms:
-5.75 + L = 0.25

Solving
-5.75 + L = 0.25

Solving for variable 'L'.

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

Add '5.75' to each side of the equation.
-5.75 + 5.75 + L = 0.25 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + L = 0.25 + 5.75
L = 0.25 + 5.75

Combine like terms: 0.25 + 5.75 = 6
L = 6

Simplifying
L = 6


Subproblem 2
L + -5.75 = -0.25

Simplifying
L + -5.75 = -0.25

Reorder the terms:
-5.75 + L = -0.25

Solving
-5.75 + L = -0.25

Solving for variable 'L'.

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

Add '5.75' to each side of the equation.
-5.75 + 5.75 + L = -0.25 + 5.75

Combine like terms: -5.75 + 5.75 = 0.00
0.00 + L = -0.25 + 5.75
L = -0.25 + 5.75

Combine like terms: -0.25 + 5.75 = 5.5
L = 5.5

Simplifying
L = 5.5


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {6, 5.5}

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