L*(2L+1)=32

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Solution for L*(2L+1)=32 equation: 


Simplifying
L(2L + 1) = 32

Reorder the terms:
L(1 + 2L) = 32
(1 * L + 2L * L) = 32
(1L + 2L2) = 32

Solving
1L + 2L2 = 32

Solving for variable 'L'.

Reorder the terms:
-32 + 1L + 2L2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + 1L + 2L2 = 0

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

Divide each side by '2'.
-16 + 0.5L + L2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 0.5L + 16 + L2 = 0 + 16

Reorder the terms:
-16 + 16 + 0.5L + L2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 0.5L + L2 = 0 + 16
0.5L + L2 = 0 + 16

Combine like terms: 0 + 16 = 16
0.5L + L2 = 16

The L term is 0.5L.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5L + 0.0625 + L2 = 16 + 0.0625

Reorder the terms:
0.0625 + 0.5L + L2 = 16 + 0.0625

Combine like terms: 16 + 0.0625 = 16.0625
0.0625 + 0.5L + L2 = 16.0625

Factor a perfect square on the left side:
(L + 0.25)(L + 0.25) = 16.0625

Calculate the square root of the right side: 4.007804885

Break this problem into two subproblems by setting 
(L + 0.25) equal to 4.007804885 and -4.007804885.


Subproblem 1
L + 0.25 = 4.007804885

Simplifying
L + 0.25 = 4.007804885

Reorder the terms:
0.25 + L = 4.007804885

Solving
0.25 + L = 4.007804885

Solving for variable 'L'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + L = 4.007804885 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + L = 4.007804885 + -0.25
L = 4.007804885 + -0.25

Combine like terms: 4.007804885 + -0.25 = 3.757804885
L = 3.757804885

Simplifying
L = 3.757804885


Subproblem 2
L + 0.25 = -4.007804885

Simplifying
L + 0.25 = -4.007804885

Reorder the terms:
0.25 + L = -4.007804885

Solving
0.25 + L = -4.007804885

Solving for variable 'L'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + L = -4.007804885 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + L = -4.007804885 + -0.25
L = -4.007804885 + -0.25

Combine like terms: -4.007804885 + -0.25 = -4.257804885
L = -4.257804885

Simplifying
L = -4.257804885


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {3.757804885, -4.257804885}

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