2/3x=x+2

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

2/3x=x+2

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R


We get rid of parentheses

2/3x-x-2=0

We multiply all the terms by the denominator


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

Wy multiply elements

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

a = -3; b = -6; c = +2;
Δ = b2-4ac
Δ = -62-4·(-3)·2
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{15}}{2*-3}=\frac{6-2\sqrt{15}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{15}}{2*-3}=\frac{6+2\sqrt{15}}{-6}
$