Answer:
-1, 2, 6
Step-by-step explanation:
We have to solve the equation as follows: 1/(x-6) + (x/(x-2)) = (4/(x²-8x+12)).
Now, we have, [tex]\frac{1}{x-6} +\frac{x}{x-2} = \frac{4}{x^{2}-8x+12 }[/tex]
⇒[tex]\frac{(x-2)+x(x-6)}{(x-2)(x-6)} = \frac{4}{x^{2}-8x+12 }[/tex]
⇒[tex]\frac{x-2+x^{2}-6x }{(x-2)(x-6)} =\frac{4}{(x-2)(x-6)}[/tex]
⇒[tex]\frac{(x-2)(x-6)}{x^{2}-5x-2 }=\frac{(x-2)(x-6)}{4}[/tex]
⇒[tex](x-2)(x-6)[\frac{1}{x^{2} -5x-2} -\frac{1}{4} ]=0[/tex]
⇒ [tex](x-2)(x-6) =0[/tex] or, [tex][\frac{1}{x^{2} -5x-2} -\frac{1}{4} ]=0[/tex]
If, (x-2)(x-6) =0, then x=2 or x=6
If, [tex][\frac{1}{x^{2} -5x-2} -\frac{1}{4} ]=0[/tex], then [tex]x^{2} -5x-2=4[/tex]
and (x-6)(x+1) =0
Therefore, x=6 or -1
So the solutions for x are -1, 2 6. (Answer)
