L^2-8L-90=0

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Solution for L^2-8L-90=0 equation: 


Simplifying
L2 + -8L + -90 = 0

Reorder the terms:
-90 + -8L + L2 = 0

Solving
-90 + -8L + L2 = 0

Solving for variable 'L'.

Begin completing the square.

Move the constant term to the right:

Add '90' to each side of the equation.
-90 + -8L + 90 + L2 = 0 + 90

Reorder the terms:
-90 + 90 + -8L + L2 = 0 + 90

Combine like terms: -90 + 90 = 0
0 + -8L + L2 = 0 + 90
-8L + L2 = 0 + 90

Combine like terms: 0 + 90 = 90
-8L + L2 = 90

The L term is -8L.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8L + 16 + L2 = 90 + 16

Reorder the terms:
16 + -8L + L2 = 90 + 16

Combine like terms: 90 + 16 = 106
16 + -8L + L2 = 106

Factor a perfect square on the left side:
(L + -4)(L + -4) = 106

Calculate the square root of the right side: 10.295630141

Break this problem into two subproblems by setting 
(L + -4) equal to 10.295630141 and -10.295630141.


Subproblem 1
L + -4 = 10.295630141

Simplifying
L + -4 = 10.295630141

Reorder the terms:
-4 + L = 10.295630141

Solving
-4 + L = 10.295630141

Solving for variable 'L'.

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

Add '4' to each side of the equation.
-4 + 4 + L = 10.295630141 + 4

Combine like terms: -4 + 4 = 0
0 + L = 10.295630141 + 4
L = 10.295630141 + 4

Combine like terms: 10.295630141 + 4 = 14.295630141
L = 14.295630141

Simplifying
L = 14.295630141


Subproblem 2
L + -4 = -10.295630141

Simplifying
L + -4 = -10.295630141

Reorder the terms:
-4 + L = -10.295630141

Solving
-4 + L = -10.295630141

Solving for variable 'L'.

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

Add '4' to each side of the equation.
-4 + 4 + L = -10.295630141 + 4

Combine like terms: -4 + 4 = 0
0 + L = -10.295630141 + 4
L = -10.295630141 + 4

Combine like terms: -10.295630141 + 4 = -6.295630141
L = -6.295630141

Simplifying
L = -6.295630141


Solution
The solution to the problem is based on the solutions
from the subproblems.
L = {14.295630141, -6.295630141}

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