Question

`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3,x≠2,4` 

 

Answer 1

`(x-1)/(x-2)+(x-3)/(x-4)=3 1/3,x≠2,4` 

⇒`((x-1)(x-4)+(x-2)(x-3))/((x-2)(x-4))0=10/3` 

⇒`(x^2-5x+4+x^2-5x+6)/(x^2-6x+8)=10/3`  

⇒`(2x^2-10x+10)/(x^2-6x+8)=10/3`  

⇒`(x^2-5x+5)/(x^2-6x+8)=5/3` 

⇒`3x^2-15x+15=5x^2-30x+40`

⇒`2x^2-15x+25=0` 

⇒`2x^2-10x-5x+25=0` 

⇒`2x(x-5)-5 (x-5)=0` 

⇒`(x-5)(2x-5)=0` 

⇒`x-5=0 or  2x-5=0` 

⇒`x=5  or  x=5/2` 

Hence, 5 and `5/2` are the roots of the given equation.