x+1/3x-2=5x-4/3x+2

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

x+1/3x-2=5x-4/3x+2

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x+2)!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

x*3x-5x*3x-2*3x-2*3x+5=0

Wy multiply elements

3x^2-15x^2-6x-6x+5=0

We add all the numbers together, and all the variables

-12x^2-12x+5=0

a = -12; b = -12; c = +5;
Δ = b2-4ac
Δ = -122-4·(-12)·5
Δ = 384
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:

$\sqrt{\Delta}=\sqrt{384}=\sqrt{64*6}=\sqrt{64}*\sqrt{6}=8\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-8\sqrt{6}}{2*-12}=\frac{12-8\sqrt{6}}{-24}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+8\sqrt{6}}{2*-12}=\frac{12+8\sqrt{6}}{-24}
$