(1+0.25l)l=24

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Solution for (1+0.25l)l=24 equation: 


Simplifying
(1 + 0.25l) * l = 24

Reorder the terms for easier multiplication:
l(1 + 0.25l) = 24
(1 * l + 0.25l * l) = 24
(1l + 0.25l2) = 24

Solving
1l + 0.25l2 = 24

Solving for variable 'l'.

Reorder the terms:
-24 + 1l + 0.25l2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + 1l + 0.25l2 = 0

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

Divide each side by '0.25'.
-96 + 4l + l2 = 0

Move the constant term to the right:

Add '96' to each side of the equation.
-96 + 4l + 96 + l2 = 0 + 96

Reorder the terms:
-96 + 96 + 4l + l2 = 0 + 96

Combine like terms: -96 + 96 = 0
0 + 4l + l2 = 0 + 96
4l + l2 = 0 + 96

Combine like terms: 0 + 96 = 96
4l + l2 = 96

The l term is 4l.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4l + 4 + l2 = 96 + 4

Reorder the terms:
4 + 4l + l2 = 96 + 4

Combine like terms: 96 + 4 = 100
4 + 4l + l2 = 100

Factor a perfect square on the left side:
(l + 2)(l + 2) = 100

Calculate the square root of the right side: 10

Break this problem into two subproblems by setting 
(l + 2) equal to 10 and -10.


Subproblem 1
l + 2 = 10

Simplifying
l + 2 = 10

Reorder the terms:
2 + l = 10

Solving
2 + l = 10

Solving for variable 'l'.

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

Add '-2' to each side of the equation.
2 + -2 + l = 10 + -2

Combine like terms: 2 + -2 = 0
0 + l = 10 + -2
l = 10 + -2

Combine like terms: 10 + -2 = 8
l = 8

Simplifying
l = 8


Subproblem 2
l + 2 = -10

Simplifying
l + 2 = -10

Reorder the terms:
2 + l = -10

Solving
2 + l = -10

Solving for variable 'l'.

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

Add '-2' to each side of the equation.
2 + -2 + l = -10 + -2

Combine like terms: 2 + -2 = 0
0 + l = -10 + -2
l = -10 + -2

Combine like terms: -10 + -2 = -12
l = -12

Simplifying
l = -12


Solution
The solution to the problem is based on the solutions
from the subproblems.
l = {8, -12}

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