3 min read Jun 17, 2024

Simplifying the Expression: (x^2-8x+12)(x^2-16)/(x^2-36)(x^2-4)

This expression involves a product of four quadratic expressions, and it can be simplified by factoring and canceling common terms. Let's break down the steps:

1. Factoring the Quadratic Expressions

  • (x^2-8x+12) can be factored as (x-6)(x-2)
  • (x^2-16) is a difference of squares, factoring as (x+4)(x-4)
  • (x^2-36) is also a difference of squares, factoring as (x+6)(x-6)
  • (x^2-4) is a difference of squares, factoring as (x+2)(x-2)

2. Substituting the Factored Expressions

Now, substitute these factored expressions back into the original expression:

[(x-6)(x-2)(x+4)(x-4)] / [(x+6)(x-6)(x+2)(x-2)]

3. Canceling Common Terms

Observe that several terms appear in both the numerator and denominator. We can cancel these common terms:

( (x-6)(x-2)(x+4)(x-4) ) / ( (x+6)( (x-6) )(x+2)( (x-2) )

This leaves us with:

(x+4)(x-4) / (x+6)(x+2)

4. Final Simplified Expression

The simplified expression after canceling common terms is (x+4)(x-4) / (x+6)(x+2). We can leave it in this factored form, or multiply it out to get (x^2-16) / (x^2 + 8x + 12).

Important Note: This simplified expression is equivalent to the original one except for the values of x that make the original expression undefined, which are:

  • x = 6
  • x = -6
  • x = 2
  • x = -2

These values make the original expression undefined because they would lead to division by zero in the denominator.