Finding X-Intercepts: F(x) = (4x^2 - 36x) / (x - 9)

Hey guys! Let's dive into how to find the x-intercepts of the function f(x) = (4x^2 - 36x) / (x - 9). This is a classic algebra problem, and understanding how to solve it can really boost your math skills. We'll break it down step-by-step, so it’s super easy to follow. Remember, x-intercepts are the points where the graph of the function crosses the x-axis, meaning the value of f(x) is zero at these points. To find them, we need to set f(x) equal to zero and solve for x. This involves a bit of algebraic manipulation, but don't worry, we’ll take it nice and slow. By the end of this article, you’ll be a pro at finding x-intercepts for rational functions like this one! So, grab your pencils and let’s get started!

Understanding X-Intercepts

First off, let's make sure we're all on the same page about what an x-intercept actually is. In simple terms, the x-intercept is where a function's graph crosses the x-axis. Think of it like this: if you're driving along the x-axis, the x-intercept is the spot where your car (the function's graph) touches the road (the x-axis). Mathematically, this happens when f(x), which represents the y-value, is equal to zero. So, to find the x-intercepts, we're essentially looking for the x-values that make the function equal to zero. Why is this important? Well, x-intercepts can tell us a lot about a function's behavior, like where it changes sign (from positive to negative or vice versa) and where it has roots (or solutions). They're like little clues that help us understand the bigger picture of the function's graph and properties. For the given function, f(x) = (4x^2 - 36x) / (x - 9), we're specifically interested in finding the values of x that make the numerator equal to zero because a fraction is zero only when its numerator is zero (as long as the denominator isn't also zero at the same point – we’ll get to that in a bit!). This leads us to the next step: setting the numerator equal to zero and solving for x. Remember, finding x-intercepts is a fundamental skill in algebra and calculus, so mastering it is definitely worth the effort!

Setting the Function to Zero

Okay, now that we know what x-intercepts are, let's get down to business and find them for our function, f(x) = (4x^2 - 36x) / (x - 9). The key here is to set the function equal to zero and solve for x. Remember, a fraction is equal to zero only when its numerator is zero (and the denominator is not zero at the same x-value). So, we can focus solely on the numerator of our function. This simplifies the problem quite a bit! We'll take the numerator, 4x^2 - 36x, and set it equal to zero. This gives us the equation 4x^2 - 36x = 0. Now, our goal is to solve this equation for x. There are a couple of ways we can approach this. One way is to try and factor the quadratic expression. Factoring is a super useful technique in algebra, and it can make solving equations like this much easier. Another way is to use the quadratic formula, but in this case, factoring is probably the simpler route. Once we've factored the expression (or used another method to solve the equation), we'll have the values of x that make the numerator zero. These values are our potential x-intercepts. But remember, we need to check if these values also make the denominator zero, because if they do, they won't be valid x-intercepts (they'll be something else entirely, like a hole in the graph!). So, let’s roll up our sleeves and get to solving 4x^2 - 36x = 0.

Factoring the Numerator

Alright, let's tackle that numerator: 4x^2 - 36x. Our mission is to factor this expression, which means we want to rewrite it as a product of simpler expressions. The first thing to look for when factoring is a common factor. Do you see anything that divides evenly into both terms, 4x^2 and -36x? You got it! Both terms have a factor of 4x. So, we can factor out 4x from the expression. When we factor out 4x, we're essentially dividing each term by 4x and writing the result inside parentheses. So, 4x^2 divided by 4x is x, and -36x divided by 4x is -9. This means we can rewrite 4x^2 - 36x as 4x(x - 9). See how much simpler that looks? Now, we have our numerator in factored form. This is a huge step because it makes it much easier to find the values of x that make the expression equal to zero. Remember, our goal is to solve the equation 4x(x - 9) = 0. When we have a product of factors equal to zero, it means that at least one of the factors must be zero. This is a crucial concept for solving equations by factoring. So, let’s take each factor, set it equal to zero, and solve for x. This will give us the potential x-intercepts of our function. Factoring is a powerful tool in algebra, and this example really shows how it can simplify complex expressions and make solving equations much more manageable. Keep practicing your factoring skills, and you'll become a master in no time!

Solving for x

Now that we've factored the numerator, 4x^2 - 36x, into 4x(x - 9), we can solve for x. Remember, we set the numerator equal to zero, so we have the equation 4x(x - 9) = 0. The beauty of factoring is that it allows us to use the zero-product property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. In our case, we have two factors: 4x and (x - 9). So, to solve for x, we'll set each factor equal to zero and solve the resulting equations. First, let's take the factor 4x and set it equal to zero: 4x = 0. To solve for x, we simply divide both sides of the equation by 4. This gives us x = 0. So, one potential x-intercept is x = 0. Now, let's take the second factor, (x - 9), and set it equal to zero: x - 9 = 0. To solve for x, we add 9 to both sides of the equation. This gives us x = 9. So, another potential x-intercept is x = 9. We've found two possible x-intercepts: x = 0 and x = 9. But before we declare victory, we need to do one more crucial step: check these values in the original function to make sure they are valid x-intercepts. Remember, we need to make sure they don't also make the denominator zero, because that would mean they're not x-intercepts (they might be something else, like a vertical asymptote or a hole in the graph).

Checking for Extraneous Solutions

Okay, we've found two potential x-intercepts: x = 0 and x = 9. But hold on a second! Before we circle these as our final answers, we need to do a crucial check. We have to make sure these values don't make the denominator of our original function, f(x) = (4x^2 - 36x) / (x - 9), equal to zero. Why? Because if the denominator is zero, the function is undefined at that point. This means that the point wouldn't actually be an x-intercept, but rather something else entirely, like a vertical asymptote or a hole in the graph. So, let's plug in each of our potential x-intercepts into the denominator, (x - 9), and see what happens. First, let's try x = 0. If we plug in 0, we get 0 - 9 = -9. That's not zero, so x = 0 is still a valid x-intercept. Great! Now, let's try x = 9. If we plug in 9, we get 9 - 9 = 0. Uh oh! This means that x = 9 makes the denominator zero, so it's not a valid x-intercept. It's what we call an extraneous solution. Extraneous solutions can pop up when we're solving equations, especially rational equations like this one, where we have variables in the denominator. They're essentially solutions that we find algebraically, but they don't actually work in the original equation. So, after all that work, it turns out that only one of our potential x-intercepts is actually a real x-intercept. This highlights the importance of checking your solutions, especially when dealing with rational functions. So, what's our final answer? Let's nail it down.

The Final Answer

We've gone through all the steps: setting the function to zero, factoring the numerator, solving for x, and most importantly, checking for extraneous solutions. After all that, we've arrived at our final answer! We initially found two potential x-intercepts: x = 0 and x = 9. However, we discovered that x = 9 makes the denominator of the function zero, which means it's an extraneous solution and not a valid x-intercept. Therefore, the only x-intercept of the function f(x) = (4x^2 - 36x) / (x - 9) is x = 0. This means that the graph of the function crosses the x-axis only at the point (0, 0). It's always a good idea to think about what this means graphically. An x-intercept tells us where the function's graph intersects the x-axis. In this case, the graph crosses the x-axis at the origin. Understanding x-intercepts is a fundamental concept in algebra and calculus, and being able to find them is a valuable skill. So, congratulations on working through this problem with me! You've now got a solid understanding of how to find x-intercepts for rational functions. Keep practicing, and you'll become even more confident in your math abilities. Remember, math isn't about memorizing formulas; it's about understanding the concepts and applying them. And you've just demonstrated that you can do exactly that!