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

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

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

We move all terms to the left:

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

We get rid of parentheses

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

We calculate fractions


x-x+(-5x)/3x^2+(-15x)/3x^2+1=0

We add all the numbers together, and all the variables

(-5x)/3x^2+(-15x)/3x^2+1=0

We multiply all the terms by the denominator


(-5x)+(-15x)+1*3x^2=0

Wy multiply elements

3x^2+(-5x)+(-15x)=0

We get rid of parentheses

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

We add all the numbers together, and all the variables

3x^2-20x=0

a = 3; b = -20; c = 0;
Δ = b2-4ac
Δ = -202-4·3·0
Δ = 400
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{400}=20$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-20)-20}{2*3}=\frac{0}{6}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-20)+20}{2*3}=\frac{40}{6}
=6+2/3
$