2x/x-3=x-2/x-5

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

2x/x-3=x-2/x-5

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R



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

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


2x-x*x+5*x-3*x+2=0

We add all the numbers together, and all the variables

4x-x*x+2=0

Wy multiply elements

-1x^2+4x+2=0

a = -1; b = 4; c = +2;
Δ = b2-4ac
Δ = 42-4·(-1)·2
Δ = 24
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:

$\sqrt{\Delta}=\sqrt{24}=\sqrt{4*6}=\sqrt{4}*\sqrt{6}=2\sqrt{6}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-2\sqrt{6}}{2*-1}=\frac{-4-2\sqrt{6}}{-2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+2\sqrt{6}}{2*-1}=\frac{-4+2\sqrt{6}}{-2}
$