(x^2-x-20)/(x^2-25)

My teacher said that the first step is to factor everything. I know that x^2-25 is factored as (x-5) (x+5), but still need help with factoring x^2-x-20.

It is (x-5)(x+4)

just having the problem given the way it was, you ought to have looked for a factor in common with the numerator.

Don't let the teacher be cleverer than you!

thanks!

To factorize the expression x^2 - x - 20, we need to find two numbers that multiply to give -20 and add up to -1 (coefficient of x).

We can start by listing all the possible pairs of numbers that multiply to give -20:
1, -20
-1, 20
2, -10
-2, 10
4, -5
-4, 5

Now, we need to identify which pair adds up to -1. In this case, -4 and 5 add up to -1.

So, we can rewrite -x as -4x + 5x:

x^2 - 4x + 5x - 20

Now, we can group the terms and apply factoring by grouping:

(x^2 - 4x) + (5x - 20)

Taking out the common factor from the first two terms and the last two terms:

x(x - 4) + 5(x - 4)

Notice that we have a common binomial term (x - 4). We can factor it out:

(x - 4)(x + 5)

Therefore, the factorization of x^2 - x - 20 is (x - 4)(x + 5).

Now, we can rewrite the original expression:

(x^2 - x - 20)/(x^2 - 25) = ((x - 4)(x + 5))/((x - 5)(x + 5))

(x + 5) can be canceled out from the numerator and denominator:

(x - 4)/(x - 5)

So, the simplified expression is (x - 4)/(x - 5).