x+5/x-13=2x/x-5

Solution for x+5/x-13=2x/x-5 equation:

x+5/x-13=2x/x-5

We move all terms to the left:

x+5/x-13-(2x/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

x+5/x-2x/x+5-13=0

We multiply all the terms by the denominator


x*x-2x+5*x-13*x+5=0

We add all the numbers together, and all the variables

-10x+x*x+5=0

Wy multiply elements

x^2-10x+5=0

a = 1; b = -10; c = +5;
Δ = b2-4ac
Δ = -102-4·1·5
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-4\sqrt{5}}{2*1}=\frac{10-4\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+4\sqrt{5}}{2*1}=\frac{10+4\sqrt{5}}{2}
$