(2x-6)-1/3x=5/3x+4

Solution for (2x-6)-1/3x=5/3x+4 equation:

(2x-6)-1/3x=5/3x+4

We move all terms to the left:

(2x-6)-1/3x-(5/3x+4)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x+4)!=0

x∈R


We get rid of parentheses

2x-1/3x-5/3x-6-4=0

We multiply all the terms by the denominator


2x*3x-6*3x-4*3x-1-5=0

We add all the numbers together, and all the variables

2x*3x-6*3x-4*3x-6=0

Wy multiply elements

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

We add all the numbers together, and all the variables

6x^2-30x-6=0

a = 6; b = -30; c = -6;
Δ = b2-4ac
Δ = -302-4·6·(-6)
Δ = 1044
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:

$\sqrt{\Delta}=\sqrt{1044}=\sqrt{36*29}=\sqrt{36}*\sqrt{29}=6\sqrt{29}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-6\sqrt{29}}{2*6}=\frac{30-6\sqrt{29}}{12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+6\sqrt{29}}{2*6}=\frac{30+6\sqrt{29}}{12}
$