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