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

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

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

We move all terms to the left:

x-2/3x-(1/3x+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-2/3x-1/3x-2=0

We multiply all the terms by the denominator


x*3x-2*3x-2-1=0

We add all the numbers together, and all the variables

x*3x-2*3x-3=0

Wy multiply elements

3x^2-6x-3=0

a = 3; b = -6; c = -3;
Δ = b2-4ac
Δ = -62-4·3·(-3)
Δ = 72
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:

$\sqrt{\Delta}=\sqrt{72}=\sqrt{36*2}=\sqrt{36}*\sqrt{2}=6\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-6\sqrt{2}}{2*3}=\frac{6-6\sqrt{2}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+6\sqrt{2}}{2*3}=\frac{6+6\sqrt{2}}{6}
$