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

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

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

We move all terms to the left:

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


Domain of the equation: (2x-3)!=0

We move all terms containing x to the left, all other terms to the right

2x!=3

x!=3/2

x!=1+1/2

x∈R



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

x∈R


We get rid of parentheses

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

We calculate fractions


3x/(2x^2-3x)+(-4x+6)/(2x^2-3x)-5=0

We multiply all the terms by the denominator


3x+(-4x+6)-5*(2x^2-3x)=0

We multiply parentheses

-10x^2+3x+(-4x+6)+15x=0

We get rid of parentheses

-10x^2+3x-4x+15x+6=0

We add all the numbers together, and all the variables

-10x^2+14x+6=0

a = -10; b = 14; c = +6;
Δ = b2-4ac
Δ = 142-4·(-10)·6
Δ = 436
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{436}=\sqrt{4*109}=\sqrt{4}*\sqrt{109}=2\sqrt{109}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{109}}{2*-10}=\frac{-14-2\sqrt{109}}{-20}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{109}}{2*-10}=\frac{-14+2\sqrt{109}}{-20}
$