Factoring Quadratics And Finding Zeroes F(x) = X^2 - 10x + 25

Hey guys! Let's dive deep into the world of quadratic functions. Specifically, we're going to break down how to find the zeroes and factored form of a function. This is super useful stuff, whether you're tackling algebra problems or just want to understand how equations work. Our example today is f(x) = x^2 - 10x + 25. So, grab your thinking caps, and let's get started!

Understanding Quadratic Functions and Zeroes

Before we jump into the nitty-gritty, let's make sure we're all on the same page. A quadratic function is a polynomial function of degree two. That basically means the highest power of 'x' in the equation is 2. Our example, f(x) = x^2 - 10x + 25, totally fits the bill. These functions create a U-shaped curve when graphed, known as a parabola. Understanding the zeroes, also known as roots or x-intercepts, is crucial in analyzing these parabolas. The zeroes of a function are the values of 'x' that make the function equal to zero. In other words, they are the points where the parabola intersects the x-axis. Finding these zeroes helps us understand the behavior and key features of the quadratic function. There are several methods to find the zeroes of a quadratic function, such as factoring, using the quadratic formula, or completing the square. Factoring is often the easiest method when the quadratic expression can be factored neatly. The quadratic formula is a universal method that works for any quadratic equation, while completing the square is a method that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. Knowing these zeroes is like having a map to the function's behavior – it tells us where the graph crosses the x-axis, which is super helpful for sketching the parabola and solving related problems. Plus, finding zeroes is a fundamental skill in algebra and calculus, so mastering it now will definitely pay off later.

Factoring the Quadratic Function f(x) = x^2 - 10x + 25

Now, let's roll up our sleeves and get to the fun part: factoring! Our function is f(x) = x^2 - 10x + 25. Factoring is like reverse multiplication – we're trying to find two expressions that multiply together to give us our original quadratic expression. When you look at x^2 - 10x + 25, you might notice something special. It looks like a perfect square trinomial. A perfect square trinomial is a quadratic expression that can be factored into the form (ax + b)^2 or (ax - b)^2. To identify a perfect square trinomial, check if the first and last terms are perfect squares and if the middle term is twice the product of the square roots of the first and last terms. In our case, x^2 is a perfect square (x * x), and 25 is a perfect square (5 * 5). The middle term, -10x, is indeed twice the product of x and -5 (2 * x * -5 = -10x). Bingo! This means we can factor f(x) = x^2 - 10x + 25 into a perfect square. Factoring perfect square trinomials is a fantastic shortcut, guys. It simplifies the process and reduces the chances of making mistakes. The factored form will be (x - 5)(x - 5), which we can also write as (x - 5)^2. See how neatly that works? Factoring isn't just about finding the right numbers; it's about recognizing patterns that make the whole process smoother and quicker. This skill is super handy not just for solving equations but also for simplifying expressions and understanding the relationships between different parts of a function.

Determining the Zero of the Function

Alright, we've factored the function – awesome! Now, let's find the zero. Remember, the zero of a function is the value of x that makes f(x) equal to zero. We've got our function in factored form: f(x) = (x - 5)^2. This makes finding the zero super straightforward. To find the zero, we set f(x) to zero: (x - 5)^2 = 0. Now, we need to solve for x. Since we have a squared term, we can take the square root of both sides of the equation. The square root of (x - 5)^2 is simply (x - 5), and the square root of 0 is 0. So, we have x - 5 = 0. To isolate x, we add 5 to both sides of the equation: x = 5. And there you have it! The zero of the function f(x) = x^2 - 10x + 25 is x = 5. But wait, there's a little more to this. Because we had (x - 5) squared, this zero has a multiplicity of 2. What does that mean? It means that the graph of the function touches the x-axis at x = 5 but doesn't cross it. Think of it like the parabola bouncing off the x-axis at that point. Understanding the concept of multiplicity helps us get a clearer picture of the function's behavior around its zeroes. So, not only have we found the zero, but we also know how the graph interacts with the x-axis at that point. Pretty cool, right?

Writing the Factored Form of f(x)

We've already touched on the factored form, but let's make it super clear. The factored form of a quadratic function is the way we write it as a product of its factors. This form is incredibly useful because it directly shows us the zeroes of the function. For our function, f(x) = x^2 - 10x + 25, we found that it factors into (x - 5)(x - 5). We can also write this as (x - 5)^2. This is the factored form of the function. Writing a function in its factored form makes it super easy to identify the zeroes. Each factor corresponds to a zero of the function. In our case, the factor (x - 5) tells us that x = 5 is a zero. The beauty of the factored form is that it gives us this information almost instantly. No need to solve equations or use the quadratic formula; the zeroes are right there in the factors. Plus, the exponent on the factor tells us about the multiplicity of the zero, as we discussed earlier. Factored form is not just a way to write the function; it's a powerful tool for understanding its behavior and quickly extracting key information. It's like having a decoder ring for quadratic functions! So, whenever you can, try to factor a quadratic function – it's a game-changer.

Putting It All Together

Okay, guys, let's recap what we've done. We started with the quadratic function f(x) = x^2 - 10x + 25. First, we recognized that this is a perfect square trinomial, which made factoring a breeze. We factored the function into f(x) = (x - 5)^2. Next, we set the factored form equal to zero to find the zeroes of the function: (x - 5)^2 = 0. Solving for x, we found that the zero is x = 5, with a multiplicity of 2. Finally, we highlighted that (x - 5)^2 is the factored form of the function. We've seen how factoring can simplify the process of finding zeroes and how the factored form gives us direct insight into the function's behavior. These skills are super useful for tackling all sorts of quadratic equations and problems. The key takeaways here are: recognize special patterns like perfect square trinomials, understand what zeroes represent on a graph, and appreciate the power of factored form. By mastering these concepts, you'll be well-equipped to handle more complex quadratic functions and their applications. Keep practicing, and you'll become a factoring pro in no time!

Conclusion

So, that's a wrap, folks! We've successfully determined the zero and the factored form of f(x) = x^2 - 10x + 25. Remember, the zero is x = 5, and the factored form is (x - 5)^2. We've seen how factoring simplifies the process and provides valuable insights into the function's behavior. Factoring and finding zeroes are fundamental skills in algebra, and mastering them will open doors to more advanced math concepts. Keep practicing, and you'll become a quadratic function whiz! Happy mathing, guys!