Question

How do you express `(x+2) / (x(x-4))` in partial fractions?

Answer 1

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

Explanation:

Here,

`(1)(x+2)/(x(x-4))=1/2[(2x+4)/(x(x-4))]`

`color(white)((x+2)/(x(x-4)))=1/2[(3x-(x-4))/(x(x-4))]`

`color(white)((x+2)/(x(x-4)))=1/2[(3x)/(x(x-4))-(x-4)/(x(x-4))]`

`color(white)((x+2)/(x(x-4)))=1/2[3/(x-4)-1/x]`

`color(white)((x+2)/(x(x-4)))=3/(2(x-4))-1/(2x)`

`(2)(x+2)/(x(x-4))=A/(x-4)+B/x`

`=>x+2=Ax+B(x-4)`

Take ,

`x=4=>4+2=A(4)=>4A=6=>A=3/2`

`x=0=>0+2=B(-4)=>B=-2/4=-1/2`

Subst. values of `A and B` into `(2)`

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

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