Question

How do you simplify `(x^3 - 2x^2 - 3x)/(2x^2 - 6x)`?

Answer 1

You can start by factoring out what's multiplying all the terms both in the numerator and denominator; then, we can find the roots of what's left and rewritethe fraction as a quotient of factors.

`(cancelx(x^2-2x-3))/(cancelx(2x-6))`

Now, let's find the roots of the quadratic:

`(2+-sqrt(4-4(1)(-3)))/2`
`(2+-4)/2`

`x_1=3`, which, equaled to zero, turns into the factor `(x-3)=0`
`x_2-1`, which, equaled to zero, turns into the factor `(x+1)=0`

And about the denominator, we have that `2` is multiplying both terms, so `(2x-6)=2(x-3)`

Now, rewriting it all over again:

`(cancel(x-3)(x+1))/(2cancel(x-3))`

Thus, the final answer is `(x+1)/2`