Question

Solve the following equation. Remember to check for extraneous solutions 1/(x-6) + (x/(x-2)) = (4/(x²-8x+12)).

Answer 1

-1, 2, 6

Explanation:

We have to solve the equation as follows: 1/(x-6) + (x/(x-2)) = (4/(x²-8x+12)).

Now, we have,
`(1)/(x-6) +(x)/(x-2) = (4)/(x^(2)-8x+12 )`

⇒
`((x-2)+x(x-6))/((x-2)(x-6)) = (4)/(x^(2)-8x+12 )`

⇒
`(x-2+x^(2)-6x )/((x-2)(x-6)) =(4)/((x-2)(x-6))`

⇒
`((x-2)(x-6))/(x^(2)-5x-2 )=((x-2)(x-6))/(4)`

⇒
`(x-2)(x-6)[(1)/(x^(2) -5x-2) -(1)/(4) ]=0`

⇒
`(x-2)(x-6) =0` or,
`[(1)/(x^(2) -5x-2) -(1)/(4) ]=0`

If, (x-2)(x-6) =0, then x=2 or x=6

If,
`[(1)/(x^(2) -5x-2) -(1)/(4) ]=0`, then
`x^(2) -5x-2=4`

and (x-6)(x+1) =0

Therefore, x=6 or -1

So the solutions for x are -1, 2 6. (Answer)