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

Solution for 4/x-1+6/3x+1=18/3x+1 equation:

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

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R



Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x+1)!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We calculate fractions


12x/3x^2+(-18x+6)/3x^2-1=0

We multiply all the terms by the denominator


12x+(-18x+6)-1*3x^2=0

Wy multiply elements

-3x^2+12x+(-18x+6)=0

We get rid of parentheses

-3x^2+12x-18x+6=0

We add all the numbers together, and all the variables

-3x^2-6x+6=0

a = -3; b = -6; c = +6;
Δ = b2-4ac
Δ = -62-4·(-3)·6
Δ = 108
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6\sqrt{3}}{2*-3}=\frac{6-6\sqrt{3}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6\sqrt{3}}{2*-3}=\frac{6+6\sqrt{3}}{-6}
$