Question

How do you simplify `\frac { \frac { 1} { 3x ^ { 2} - 3} } { \frac { 5} { x + 1} - \frac { x + 4} { x ^ { 2} - 3x - 4} }`?

Answer 1

`(1/(3x^2-3))/(5/(x+1)-(x+4)/((x+1)(x-4)))=(x-4)/(12x^2-84x+72)`

Explanation:

We can factorize `x^2-3x-4` by splitting middle term as

`x^2-4x+x-4=x(x-4)+1(x-4)=(x+1)(x-4)`

Therefore the denominator is `5/(x+1)-(x+4)/((x+1)(x-4))`

= `(5(x-4)-(x+4))/((x+1)(x-4))`

= `(4x-24)/((x+1)(x-4))`

and numerator is `1/(3x^2-3)=1/(3(x^2-1))=1/(3(x-1)(x+1))`

Hence `(1/(3x^2-3))/(5/(x+1)-(x+4)/((x+1)(x-4)))`

= `(1/(3(x-1)(x+1)))/((4x-24)/((x+1)(x-4)))`

= `1/(3(x-1)(x+1))xx((x+1)(x-4))/((4x-24)`

= `1/(3(x-1)cancel((x+1)))xx(cancel((x+1))(x-4))/(4(x-6)`

= `(x-4)/(12(x-1)(x-6))`

= `(x-4)/(12(x^2-7x+6))`

= `(x-4)/(12x^2-84x+72)`