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

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

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

We move all terms to the left:

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


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

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

x!=5

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

We multiply parentheses

-2x^2+(x-2)+(x-3)+10x+5x-25=0

We get rid of parentheses

-2x^2+x+x+10x+5x-2-3-25=0

We add all the numbers together, and all the variables

-2x^2+17x-30=0

a = -2; b = 17; c = -30;
Δ = b2-4ac
Δ = 172-4·(-2)·(-30)
Δ = 49
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{49}=7$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(17)-7}{2*-2}=\frac{-24}{-4}
=+6
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(17)+7}{2*-2}=\frac{-10}{-4}
=2+1/2
$