Simplify the expression below completely

( x^2-x-30)/( x^2-25)

Simplify the expression below completely

( x^2-x-30)/( x^2-25)

factor top and bottom

= (x-6)(x+5)/( (x-5)(x+5) )
= (x-6)/(x-5) , x ≠ -5

To simplify the given expression, we need to factorize the numerator and denominator and then see if any common factors can be canceled out.

First, let's factorize the numerator and denominator:

Numerator:
x^2 - x - 30

To factorize the numerator, we need to find two numbers that add up to -1 (coefficient of x) and multiply to give -30 (constant term). The numbers that satisfy this condition are -6 and 5:
(x - 6)(x + 5)

Denominator:
x^2 - 25

The denominator is in the form of a difference of squares, which can be factored as follows:
(x - 5)(x + 5)

Now, we rewrite the given expression using the factored forms:

(x^2 - x - 30)/(x^2 - 25) = (x - 6)(x + 5)/((x - 5)(x + 5))

Notice that the (x + 5) factor is common both in the numerator and denominator. This means we can cancel it out:

(x - 6)(x + 5)/((x - 5)(x + 5)) = (x - 6)/(x - 5)

Therefore, the simplified form of the given expression is (x - 6)/(x - 5).