Simplify (x^2 - 7x + 12) / (x^2 - 6x + 9) & Excluded Values

Hey guys! Today, we're diving into the world of rational expressions and tackling a common problem in algebra: simplifying these expressions and figuring out which values we need to exclude from their domains. Specifically, we're going to break down the expression (x^2 - 7x + 12) / (x^2 - 6x + 9). It might look a bit intimidating at first, but trust me, with a few simple steps, it becomes much easier to handle. We'll not only find the simplest form of this expression but also identify those sneaky x values that would make the denominator zero, which we definitely want to avoid.

So, why is this important? Well, simplifying rational expressions is a fundamental skill in algebra and calculus. It helps us solve equations, graph functions, and understand the behavior of these expressions. Plus, knowing the excluded values is crucial because it tells us where the function is undefined, preventing us from making mathematical mistakes. Think of it like this: we're not just crunching numbers; we're learning how to navigate the landscape of mathematical expressions with confidence. We'll cover everything from factoring quadratics to identifying those pesky excluded values, making sure you've got a solid grasp on the process. Let's jump right in and make some math magic happen!

Step 1: Factoring the Numerator and Denominator

The first key to simplifying any rational expression is factoring. We need to break down both the numerator and the denominator into their simplest factors. This involves looking for common factors, using factoring techniques for quadratics, and sometimes even recognizing special patterns like the difference of squares. Factoring is like taking a complex puzzle and breaking it into smaller, manageable pieces. It allows us to see the underlying structure of the expression and identify opportunities for simplification.

Let's start with the numerator, which is the quadratic expression x^2 - 7x + 12. We need to find two numbers that multiply to 12 and add up to -7. After a little thought, we can see that -3 and -4 fit the bill perfectly because (-3) * (-4) = 12 and (-3) + (-4) = -7. So, we can factor the numerator as (x - 3)(x - 4). Remember, factoring is all about reversing the process of expanding brackets, so it's a good idea to mentally check if our factored form multiplies back to the original expression.

Now, let's move on to the denominator, which is x^2 - 6x + 9. This looks like a perfect square trinomial, which is a special pattern that can make factoring even easier. A perfect square trinomial takes the form (a^2 - 2ab + b^2) or (a^2 + 2ab + b^2), and it can be factored as (a - b)^2 or (a + b)^2, respectively. In our case, we can rewrite the denominator as x^2 - 2(3)(x) + 3^2, which clearly fits the pattern of a perfect square trinomial. Therefore, we can factor it as (x - 3)^2, which is the same as (x - 3)(x - 3). Factoring can sometimes feel like detective work, but with practice, you'll start to recognize these patterns more easily. Now that we've factored both the numerator and the denominator, we're ready for the next step: simplifying the expression.

Step 2: Simplifying the Expression

With both the numerator and the denominator factored, we can now simplify the rational expression. This is where we look for common factors that appear in both the top and the bottom of the fraction. Think of it like canceling out matching pieces in a puzzle – it helps us reduce the expression to its simplest form. Simplifying not only makes the expression easier to work with but also gives us a clearer picture of its underlying behavior.

We have our expression as ((x - 3)(x - 4)) / ((x - 3)(x - 3)). Notice that the factor (x - 3) appears in both the numerator and the denominator. This means we can cancel out one (x - 3) from each. Canceling common factors is a fundamental step in simplifying rational expressions, and it's based on the principle that dividing both the numerator and denominator by the same non-zero quantity doesn't change the value of the expression.

After canceling the common factor, we're left with (x - 4) / (x - 3). This is the simplified form of our original expression. It's much cleaner and easier to work with, right? But remember, simplifying rational expressions isn't just about canceling factors; it's also about keeping track of any restrictions on the variable x. We'll get to that in the next step when we discuss excluded values. For now, let's appreciate the progress we've made in simplifying the expression – we've taken it from a somewhat complex form to a much more manageable one.

Step 3: Identifying Excluded Values

Now comes a crucial step in working with rational expressions: identifying the excluded values. These are the values of x that would make the denominator of the original expression equal to zero. Why do we care about this? Well, division by zero is undefined in mathematics, so any value of x that makes the denominator zero is off-limits. It's like a red flag warning us that the expression is not valid for those particular values of x.

To find the excluded values, we go back to the original denominator, x^2 - 6x + 9, or its factored form, (x - 3)(x - 3). We need to determine what value(s) of x would make this expression equal to zero. Setting (x - 3) equal to zero gives us x - 3 = 0, which we can solve for x by adding 3 to both sides. This gives us x = 3. So, 3 is an excluded value because if we plug it into the original denominator, we get (3 - 3)(3 - 3) = 0 * 0 = 0, which is not allowed.

It's important to note that we look at the original denominator, not the simplified one, when identifying excluded values. This is because simplifying the expression might hide the fact that certain values were originally problematic. In our case, even though the simplified expression is (x - 4) / (x - 3), the excluded value x = 3 is still relevant because it would have made the original denominator zero. So, identifying excluded values is like doing a background check on our expression to make sure we're not running into any mathematical trouble down the road. We've now simplified the expression and identified its excluded values, giving us a complete understanding of this rational expression.

Conclusion

Alright, guys, we've successfully navigated the world of rational expressions! We took the expression (x^2 - 7x + 12) / (x^2 - 6x + 9), factored it, simplified it to (x - 4) / (x - 3), and identified the excluded value as x = 3. This journey highlights the key steps in working with rational expressions: factoring, simplifying, and, most importantly, identifying those sneaky excluded values. These steps are not just about getting the right answer; they're about building a solid understanding of how these expressions behave.

Remember, factoring is like unlocking a secret code, simplifying is like streamlining a process, and finding excluded values is like ensuring safety. Each step is crucial in its own right, and together, they empower us to tackle more complex algebraic problems with confidence. Whether you're solving equations, graphing functions, or delving into calculus, these skills will serve you well. So, keep practicing, keep exploring, and remember that every mathematical challenge is an opportunity to learn and grow. You've got this!