18/3x-3+1/3=6/x-1

Solution for 18/3x-3+1/3=6/x-1 equation:

18/3x-3+1/3=6/x-1

We move all terms to the left:

18/3x-3+1/3-(6/x-1)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



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

x∈R


We get rid of parentheses

18/3x-6/x+1-3+1/3=0

We calculate fractions


18x/27x^2+(-162x)/27x^2+x/27x^2+1-3=0

We add all the numbers together, and all the variables

18x/27x^2+(-162x)/27x^2+x/27x^2-2=0

We multiply all the terms by the denominator


18x+(-162x)+x-2*27x^2=0

We add all the numbers together, and all the variables

19x+(-162x)-2*27x^2=0

Wy multiply elements

-54x^2+19x+(-162x)=0

We get rid of parentheses

-54x^2+19x-162x=0

We add all the numbers together, and all the variables

-54x^2-143x=0

a = -54; b = -143; c = 0;
Δ = b2-4ac
Δ = -1432-4·(-54)·0
Δ = 20449
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{20449}=143$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-143)-143}{2*-54}=\frac{0}{-108}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-143)+143}{2*-54}=\frac{286}{-108}
=-2+35/54
$