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

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

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

We move all terms to the left:

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


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x)!=0

x!=0/1

x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We calculate fractions


x-x+(-2x)/3x^2+(-9x)/3x^2+1=0

We add all the numbers together, and all the variables

(-2x)/3x^2+(-9x)/3x^2+1=0

We multiply all the terms by the denominator


(-2x)+(-9x)+1*3x^2=0

Wy multiply elements

3x^2+(-2x)+(-9x)=0

We get rid of parentheses

3x^2-2x-9x=0

We add all the numbers together, and all the variables

3x^2-11x=0

a = 3; b = -11; c = 0;
Δ = b2-4ac
Δ = -112-4·3·0
Δ = 121
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{121}=11$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-11)-11}{2*3}=\frac{0}{6}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-11)+11}{2*3}=\frac{22}{6}
=3+2/3
$