3/x-3=9/5x+6

Solution for 3/x-3=9/5x+6 equation:

3/x-3=9/5x+6

We move all terms to the left:

3/x-3-(9/5x+6)=0


Domain of the equation: x!=0

x∈R



Domain of the equation: 5x+6)!=0

x∈R


We get rid of parentheses

3/x-9/5x-6-3=0

We calculate fractions


15x/5x^2+(-9x)/5x^2-6-3=0

We add all the numbers together, and all the variables

15x/5x^2+(-9x)/5x^2-9=0

We multiply all the terms by the denominator


15x+(-9x)-9*5x^2=0

Wy multiply elements

-45x^2+15x+(-9x)=0

We get rid of parentheses

-45x^2+15x-9x=0

We add all the numbers together, and all the variables

-45x^2+6x=0

a = -45; b = 6; c = 0;
Δ = b2-4ac
Δ = 62-4·(-45)·0
Δ = 36
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{36}=6$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-6}{2*-45}=\frac{-12}{-90}
=2/15
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+6}{2*-45}=\frac{0}{-90}
=0
$