`(2x-3)/(x-1)^2`
Let`(2x-3)/(x-1)^2=A/(x-1)+B/(x-1)^2`
`(2x-3)/(x-1)^2=(A(x-1)+B)/(x-1)^2`
`(2x-3)/(x-1)^2=(Ax-A+B)/(x-1)^2`
`:.(2x-3)=Ax-A+B`
comparing the coefficients of the like terms,
`A=2`
`-A+B=-3`
Plug in the value of A in the above equation,
`-2+B=-3`
`B=-3+2`
`B=-1`
`:.(2x-3)/(x-1)^2=2/(x-1)-1/(x-1)^2`
Now let's check the above result,
RHS=`2/(x-1)-1/(x-1)^2`
`=(2(x-1)-1)/(x-1)^2`
`=(2x-2-1)/(x-1)^2`
`=(2x-3)/(x-1)^2`
`= LHS`
Hence it is verified.
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