l=1+1/4l

Solution for l=1+1/4l equation:

l=1+1/4l

We move all terms to the left:

l-(1+1/4l)=0


Domain of the equation: 4l)!=0

l!=0/1

l!=0

l∈R


We add all the numbers together, and all the variables

l-(1/4l+1)=0

We get rid of parentheses

l-1/4l-1=0

We multiply all the terms by the denominator


l*4l-1*4l-1=0

Wy multiply elements

4l^2-4l-1=0

a = 4; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·4·(-1)
Δ = 32
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$l_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$l_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$l_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-4\sqrt{2}}{2*4}=\frac{4-4\sqrt{2}}{8}
$
$l_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{2}}{2*4}=\frac{4+4\sqrt{2}}{8}
$