L(L+3.6)=20.77

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Solution for L(L+3.6)=20.77 equation: 


Simplifying
L(L + 3.6) = 20.77

Reorder the terms:
L(3.6 + L) = 20.77
(3.6 * L + L * L) = 20.77
(3.6L + L2) = 20.77

Solving
3.6L + L2 = 20.77

Solving for variable 'L'.

Reorder the terms:
-20.77 + 3.6L + L2 = 20.77 + -20.77

Combine like terms: 20.77 + -20.77 = 0.00
-20.77 + 3.6L + L2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '20.77' to each side of the equation.
-20.77 + 3.6L + 20.77 + L2 = 0.00 + 20.77

Reorder the terms:
-20.77 + 20.77 + 3.6L + L2 = 0.00 + 20.77

Combine like terms: -20.77 + 20.77 = 0.00
0.00 + 3.6L + L2 = 0.00 + 20.77
3.6L + L2 = 0.00 + 20.77

Combine like terms: 0.00 + 20.77 = 20.77
3.6L + L2 = 20.77

The L term is 3.6L.  Take half its coefficient (1.8).
Square it (3.24) and add it to both sides.

Add '3.24' to each side of the equation.
3.6L + 3.24 + L2 = 20.77 + 3.24

Reorder the terms:
3.24 + 3.6L + L2 = 20.77 + 3.24

Combine like terms: 20.77 + 3.24 = 24.01
3.24 + 3.6L + L2 = 24.01

Factor a perfect square on the left side:
(L + 1.8)(L + 1.8) = 24.01

Calculate the square root of the right side: 4.9

Break this problem into two subproblems by setting 
(L + 1.8) equal to 4.9 and -4.9.


Subproblem 1
L + 1.8 = 4.9

Simplifying
L + 1.8 = 4.9

Reorder the terms:
1.8 + L = 4.9

Solving
1.8 + L = 4.9

Solving for variable 'L'.

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

Add '-1.8' to each side of the equation.
1.8 + -1.8 + L = 4.9 + -1.8

Combine like terms: 1.8 + -1.8 = 0.0
0.0 + L = 4.9 + -1.8
L = 4.9 + -1.8

Combine like terms: 4.9 + -1.8 = 3.1
L = 3.1

Simplifying
L = 3.1


Subproblem 2
L + 1.8 = -4.9

Simplifying
L + 1.8 = -4.9

Reorder the terms:
1.8 + L = -4.9

Solving
1.8 + L = -4.9

Solving for variable 'L'.

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

Add '-1.8' to each side of the equation.
1.8 + -1.8 + L = -4.9 + -1.8

Combine like terms: 1.8 + -1.8 = 0.0
0.0 + L = -4.9 + -1.8
L = -4.9 + -1.8

Combine like terms: -4.9 + -1.8 = -6.7
L = -6.7

Simplifying
L = -6.7


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {3.1, -6.7}

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