2/3x=6/x+8

Solution for 2/3x=6/x+8 equation:

2/3x=6/x+8

We move all terms to the left:

2/3x-(6/x+8)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x+8)!=0

x∈R


We get rid of parentheses

2/3x-6/x-8=0

We calculate fractions


2x/3x^2+(-18x)/3x^2-8=0

We multiply all the terms by the denominator


2x+(-18x)-8*3x^2=0

Wy multiply elements

-24x^2+2x+(-18x)=0

We get rid of parentheses

-24x^2+2x-18x=0

We add all the numbers together, and all the variables

-24x^2-16x=0

a = -24; b = -16; c = 0;
Δ = b2-4ac
Δ = -162-4·(-24)·0
Δ = 256
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{256}=16$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-16)-16}{2*-24}=\frac{0}{-48}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-16)+16}{2*-24}=\frac{32}{-48}
=-2/3
$