20/4x-4+1/4=5/x-1

Solution for 20/4x-4+1/4=5/x-1 equation:

20/4x-4+1/4=5/x-1

We move all terms to the left:

20/4x-4+1/4-(5/x-1)=0


Domain of the equation: 4x!=0

x!=0/4

x!=0

x∈R



Domain of the equation: x-1)!=0

x∈R


We get rid of parentheses

20/4x-5/x+1-4+1/4=0

We calculate fractions


20x/64x^2+(-320x)/64x^2+x/64x^2+1-4=0

We add all the numbers together, and all the variables

20x/64x^2+(-320x)/64x^2+x/64x^2-3=0

We multiply all the terms by the denominator


20x+(-320x)+x-3*64x^2=0

We add all the numbers together, and all the variables

21x+(-320x)-3*64x^2=0

Wy multiply elements

-192x^2+21x+(-320x)=0

We get rid of parentheses

-192x^2+21x-320x=0

We add all the numbers together, and all the variables

-192x^2-299x=0

a = -192; b = -299; c = 0;
Δ = b2-4ac
Δ = -2992-4·(-192)·0
Δ = 89401
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{89401}=299$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-299)-299}{2*-192}=\frac{0}{-384}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-299)+299}{2*-192}=\frac{598}{-384}
=-1+107/192
$