Answer:
The lowest form of the fraction is [tex](\frac{x-3}{x-2} )^{2}[/tex]
Step-by-step explanation:
Here, the given equation is
[tex]\frac{(x-3)}{(x^{2} -4)} \times \frac{(x^{2} - x - 6) }{(x-2)}[/tex]
Now, by ALGEBRAIC IDENTITIES, we know that:
[tex]a^{2} - b^{2} = (a-b)(a+b)[/tex]
Here, [tex]( x^{2} -4) = (x^{2} - (2)^{2} ) = (x-2)(x+2)[/tex]
Also, [tex]x^{2} - x -6 = (x-3)(x+2)[/tex] (by splitting the middle term)
So, the given expression becomes:
[tex]\frac{(x-3)}{(x^{2} -4)} \times \frac{(x^{2} - x - 6) }{(x-2)}[/tex] = [tex]\frac{(x-3)}{(x-2)(x+ 2)} \times \frac{(x-3)(x+2) }{(x-2)}[/tex]
or, the expression becomes [tex]\frac{(x-3)}{(x-2)(x+ 2)} \times \frac{(x-3)(x+2) }{(x-2)}[/tex]
= [tex]\frac{(x-3)^{2} }{(x-2)^{2} } = (\frac{x-3}{x-2} )^{2}[/tex]
So,the lowest form of the fraction is [tex](\frac{x-3}{x-2} )^{2}[/tex]
