5x/3x-3+6/x+2=5/3

Solution for 5x/3x-3+6/x+2=5/3 equation:

5x/3x-3+6/x+2=5/3

We move all terms to the left:

5x/3x-3+6/x+2-(5/3)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

5x/3x+6/x-3+2-(+5/3)=0

We add all the numbers together, and all the variables

5x/3x+6/x-1-(+5/3)=0

We get rid of parentheses

5x/3x+6/x-1-5/3=0

We calculate fractions


5x^2/27x^2+162x/27x^2+(-5x)/27x^2-1=0

We multiply all the terms by the denominator


5x^2+162x+(-5x)-1*27x^2=0

Wy multiply elements

5x^2-27x^2+162x+(-5x)=0

We get rid of parentheses

5x^2-27x^2+162x-5x=0

We add all the numbers together, and all the variables

-22x^2+157x=0

a = -22; b = 157; c = 0;
Δ = b2-4ac
Δ = 1572-4·(-22)·0
Δ = 24649
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{24649}=157$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(157)-157}{2*-22}=\frac{-314}{-44}
=7+3/22
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(157)+157}{2*-22}=\frac{0}{-44}
=0
$