5/3x=10/x+30

Solution for 5/3x=10/x+30 equation:

5/3x=10/x+30

We move all terms to the left:

5/3x-(10/x+30)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: x+30)!=0

x∈R


We get rid of parentheses

5/3x-10/x-30=0

We calculate fractions


5x/3x^2+(-30x)/3x^2-30=0

We multiply all the terms by the denominator


5x+(-30x)-30*3x^2=0

Wy multiply elements

-90x^2+5x+(-30x)=0

We get rid of parentheses

-90x^2+5x-30x=0

We add all the numbers together, and all the variables

-90x^2-25x=0

a = -90; b = -25; c = 0;
Δ = b2-4ac
Δ = -252-4·(-90)·0
Δ = 625
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{625}=25$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-25)-25}{2*-90}=\frac{0}{-180}
=0
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-25)+25}{2*-90}=\frac{50}{-180}
=-5/18
$