Finding X-Intercepts: A Deep Dive Into $f(x) = X(5x+3)^2$

Hey math enthusiasts! Let's dive into a fun problem. We're going to figure out how many x-intercepts the function $f(x) = x(5x+3)^2$ has. This might seem a bit tricky at first, but trust me, it's totally manageable. We'll break it down step by step, so you'll understand it like a pro. Think of x-intercepts as the points where our function crosses or touches the x-axis. At these points, the value of the function, or $f(x)$, is always zero. That's the key to solving this, guys! So, let's get our hands dirty and see how we can nail this down. We're not just aiming to get the right answer; we're also going to understand why it's the right answer. Ready? Let's go!

Unraveling X-Intercepts: The Core Concept

Alright, before we get to the function, let's quickly recap what an x-intercept actually is. Imagine the x-axis as a horizontal line, like a tightrope. An x-intercept is simply where the graph of our function meets that tightrope. Mathematically, this happens when $f(x) = 0$. So, our mission is to find the values of x that make the function equal to zero. These x values are our x-intercepts. This is super important because it provides valuable information about the function's behavior. We can determine where the graph crosses the x-axis, and understand how the function behaves around these points. It's like having a map of the function's critical locations. Now that we understand the basics, let's apply this knowledge to our function $f(x) = x(5x+3)^2$. We need to find the values of x that make this entire expression equal to zero. Remember, the x-intercept is where $f(x) = 0$. So we are going to set our equation equal to zero. Let's do it!

To find the x-intercepts of the function $f(x) = x(5x+3)^2$, we need to solve the equation $f(x) = 0$. This means we need to find the values of x for which $x(5x+3)^2 = 0$. This equation is already factored for us, which makes our lives much easier. When we have a product of factors equal to zero, we know that at least one of the factors must be equal to zero. It's like saying if you multiply a bunch of numbers and the result is zero, then at least one of those numbers had to be zero. So, we'll set each factor equal to zero and solve for x. This will give us the x-intercepts. By solving for the zeros of the function, we gain a clear understanding of where the graph intersects the x-axis. These intersection points are not only significant in themselves, but also provide crucial information about the function's overall behavior and properties. Understanding x-intercepts can help you sketch the graph, determine the intervals where the function is positive or negative, and analyze the function's overall shape. So, let’s go ahead and find these x-intercepts!

Solving for X-Intercepts: A Step-by-Step Approach

Okay, guys, let's get to the nitty-gritty and find those x-intercepts. We've got our function $f(x) = x(5x+3)^2$. Remember, we need to find the values of x that make $f(x) = 0$. So, we set the function equal to zero: $x(5x+3)^2 = 0$. Now, since this is already factored, we can use the zero-product property. This property states that if the product of several factors is zero, then at least one of the factors must be zero. The beauty of this is that it simplifies our problem. We have two factors here: x and $(5x+3)^2$. Let's deal with them one at a time. First, we set the first factor equal to zero: $x = 0$. This immediately gives us our first x-intercept: $x = 0$. This means that the graph of the function crosses the x-axis at the point (0, 0). Easy peasy, right?

Now, let's look at the second factor, $(5x+3)^2$. We set this equal to zero: $(5x+3)^2 = 0$. To solve this, we take the square root of both sides, which gives us $5x + 3 = 0$. From here, we isolate x. Subtract 3 from both sides: $5x = -3$. Finally, divide by 5: $x = -3/5$. This gives us our second x-intercept: $x = -3/5$. However, since the factor $(5x+3)$ is squared, this x-intercept has a special property. When a factor is squared, it means the graph touches the x-axis at that point but doesn't cross it. It kind of bounces off. So, we've found two x-intercepts: $x = 0$ and $x = -3/5$. But remember, $x=-3/5$ is a repeated root. The fact that the factor $(5x+3)$ is squared tells us that the graph touches the x-axis at this point but doesn't cross it. This detail is important because it changes the behavior of the graph around that intercept. It's not just a point where the graph crosses; it's a turning point. So, while we have two values of x that give us zero, one of them has this special behavior. It's important to keep this in mind. It gives us a more complete understanding of the function's graph. We're not just looking for where the graph hits the x-axis; we're also looking at how it hits the x-axis. That means we have two x-intercepts: (0, 0) and (-3/5, 0). So, to summarize, the x-intercepts are x = 0 and x = -3/5. The function touches the x-axis at $x = -3/5$. Therefore, the function has two x-intercepts.

Decoding the X-Intercepts: The Final Count

Alright, we've done the math, we've crunched the numbers, and now it's time to figure out the final answer to our question: How many x-intercepts does the function $f(x) = x(5x+3)^2$ have? Well, we've found two values of x that make $f(x) = 0$. This means the graph touches or crosses the x-axis at two points. One is at x=0, and the other is at x = -3/5. Thus, the function has two x-intercepts. So, the function $f(x) = x(5x+3)^2$ has two x-intercepts. That's the final answer, folks! We've successfully navigated the process of finding x-intercepts, and we now understand how to analyze the function and determine how many x-intercepts it has. Remember, the key is to set the function equal to zero, factor it (if possible), and solve for x. The number of solutions for x gives you the number of x-intercepts. And don’t forget that if any factor is squared, the function touches the x-axis at that point. Knowing about repeated roots can give you extra clues about how the graph behaves around an x-intercept. Nice job, everyone! We've reached the end, guys.

We started with a function, and we broke it down step by step to find the x-intercepts. We learned about the importance of setting the function equal to zero and using the zero-product property. We also learned how to identify the repeated root. With this information, we could then identify the x-intercepts. Understanding x-intercepts is a fundamental skill in math. It helps you grasp the behavior of functions and visualize their graphs. We can find the x-intercepts, understand their meaning, and use them to get a better view of how the function behaves. Keep practicing, and you'll become a master of functions in no time. Keep the spirit alive, and keep exploring the amazing world of mathematics!