L^2+L-66=0

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Solution for L^2+L-66=0 equation: 


Simplifying
L2 + L + -66 = 0

Reorder the terms:
-66 + L + L2 = 0

Solving
-66 + L + L2 = 0

Solving for variable 'L'.

Begin completing the square.

Move the constant term to the right:

Add '66' to each side of the equation.
-66 + L + 66 + L2 = 0 + 66

Reorder the terms:
-66 + 66 + L + L2 = 0 + 66

Combine like terms: -66 + 66 = 0
0 + L + L2 = 0 + 66
L + L2 = 0 + 66

Combine like terms: 0 + 66 = 66
L + L2 = 66

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

Add '0.25' to each side of the equation.
L + 0.25 + L2 = 66 + 0.25

Reorder the terms:
0.25 + L + L2 = 66 + 0.25

Combine like terms: 66 + 0.25 = 66.25
0.25 + L + L2 = 66.25

Factor a perfect square on the left side:
(L + 0.5)(L + 0.5) = 66.25

Calculate the square root of the right side: 8.139410298

Break this problem into two subproblems by setting 
(L + 0.5) equal to 8.139410298 and -8.139410298.


Subproblem 1
L + 0.5 = 8.139410298

Simplifying
L + 0.5 = 8.139410298

Reorder the terms:
0.5 + L = 8.139410298

Solving
0.5 + L = 8.139410298

Solving for variable 'L'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + L = 8.139410298 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + L = 8.139410298 + -0.5
L = 8.139410298 + -0.5

Combine like terms: 8.139410298 + -0.5 = 7.639410298
L = 7.639410298

Simplifying
L = 7.639410298


Subproblem 2
L + 0.5 = -8.139410298

Simplifying
L + 0.5 = -8.139410298

Reorder the terms:
0.5 + L = -8.139410298

Solving
0.5 + L = -8.139410298

Solving for variable 'L'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + L = -8.139410298 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + L = -8.139410298 + -0.5
L = -8.139410298 + -0.5

Combine like terms: -8.139410298 + -0.5 = -8.639410298
L = -8.639410298

Simplifying
L = -8.639410298


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {7.639410298, -8.639410298}

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