Finding X-Axis Intercepts For Y=(x-5)(x^2-7x+12) A Step-by-Step Guide

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Hey guys! Today, we're diving into a super important concept in algebra: finding where a graph crosses the x-axis. Specifically, we're going to tackle the equation y = (x-5)(x^2 - 7x + 12). This might look a little intimidating at first, but trust me, we'll break it down into easy-to-follow steps. Understanding how to find x-axis intercepts, also known as roots or zeros, is crucial for graphing functions and solving equations. So, grab your pencils, and let's get started!

Understanding X-Axis Intercepts

Before we jump into the problem, let's make sure we're all on the same page about what an x-axis intercept actually is. Think of it this way: the x-axis is that horizontal line running across your graph. An x-axis intercept is simply the point where the graph of your equation crosses or touches this line. At these points, the y-value is always zero. This is a key concept to remember! When we talk about finding x-axis intercepts, we're essentially looking for the x-values that make y equal to zero. These points are also called the roots or zeros of the equation. Visualizing this can be incredibly helpful. Imagine a curve floating in space; the points where it dips down and touches the floor (the x-axis) are our intercepts. Knowing this, we can approach the problem with a clear goal in mind: to find the x-values that make the entire equation equal to zero. This involves a bit of algebraic manipulation and a dash of problem-solving, but nothing we can't handle!

Step 1 Factoring the Quadratic

Our equation is y = (x-5)(x^2 - 7x + 12). Notice that we already have one factor, (x-5). But we also have a quadratic expression, (x^2 - 7x + 12). To find all the x-axis intercepts, we need to factor this quadratic. Factoring a quadratic means rewriting it as a product of two binomials. So, we're looking for two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the x term). Let's think about the factors of 12: 1 and 12, 2 and 6, 3 and 4. Which pair adds up to -7? Ah, -3 and -4! So, we can factor the quadratic as (x - 3)(x - 4). Now, our equation looks like this: y = (x - 5)(x - 3)(x - 4). See? We've broken down the equation into a product of three simple factors. This is a major step in the right direction. Factoring allows us to easily identify the values of x that will make the entire expression equal to zero. It's like disassembling a machine to find the specific parts that control its function. Each factor now holds a key to finding one of our x-axis intercepts.

Step 2 Setting Factors to Zero

Now comes the fun part! We have our factored equation: y = (x - 5)(x - 3)(x - 4). Remember, we're looking for the x-values that make y equal to zero. The beauty of having factored the equation is that if any of these factors equals zero, the entire product will equal zero. Think about it: zero multiplied by anything is always zero. So, to find our x-axis intercepts, we simply need to set each factor equal to zero and solve for x. This gives us three simple equations:

  • x - 5 = 0
  • x - 3 = 0
  • x - 4 = 0

Each of these equations is a breeze to solve. We just add the constant term to both sides. This step is the heart of finding the solutions. By focusing on each factor individually, we transform a complex problem into a series of manageable tasks. It's like breaking a large boulder into smaller, easier-to-carry pieces. The principle we're using here is the Zero Product Property, a fundamental concept in algebra. It's a powerful tool that allows us to solve equations by focusing on their individual components. This property is the backbone of many algebraic techniques, and mastering it will significantly improve your problem-solving skills.

Step 3 Solving for X

Let's solve those equations! For x - 5 = 0, we add 5 to both sides, giving us x = 5. For x - 3 = 0, we add 3 to both sides, giving us x = 3. And for x - 4 = 0, we add 4 to both sides, giving us x = 4. So, we have three x-values: x = 5, x = 3, and x = 4. These are the x-coordinates where the graph crosses the x-axis. Remember, the y-coordinate at these points is always 0. Therefore, our x-axis intercepts are the points (5, 0), (3, 0), and (4, 0). Voila! We've found the solutions. Solving these simple equations is the culmination of our efforts. Each solution represents a specific point where the graph interacts with the x-axis, giving us valuable information about the function's behavior. These x-values are not just numbers; they are crucial indicators of the function's roots and its overall shape. By systematically solving each equation, we reveal the hidden points where the graph intersects the x-axis, providing a complete picture of the function's relationship with this critical line.

Step 4 Identifying the Correct Option

Now, let's look back at the original question. We were given four options: A. (-5, 0), B. (-3, 0), C. (4, 0), and D. (12, 0). We found that the x-axis intercepts are (5, 0), (3, 0), and (4, 0). Looking at the options, we see that (4, 0) matches one of our solutions. Therefore, the correct answer is C. (4, 0). It's always a good idea to double-check your work and make sure your answer makes sense in the context of the problem. In this case, we've methodically found the x-axis intercepts, and one of them perfectly matches the given options. This final step reinforces the accuracy of our solution and demonstrates the effectiveness of our problem-solving approach. By systematically comparing our results with the provided options, we can confidently identify the correct answer and move forward with a clear understanding of the solution process.

Why Other Options Are Incorrect

It's also helpful to understand why the other options are incorrect. Option A, (-5, 0), would be a solution if (x + 5) were a factor in our equation. But it's not. Option B, (-3, 0), would be a solution if (x + 3) were a factor. Again, it's not. Option D, (12, 0), doesn't correspond to any of the factors we found. Understanding why incorrect options are wrong can solidify your understanding of the concept. It's like understanding the rules of a game; knowing what you can't do is just as important as knowing what you can do. By analyzing the incorrect options, we gain a deeper insight into the relationship between factors, roots, and the overall equation. This process of elimination and critical analysis enhances our problem-solving skills and allows us to approach similar problems with greater confidence. Identifying why these options fail to satisfy the equation reinforces the importance of accurate factoring and equation solving.

Visualizing the Graph

To further solidify your understanding, it's always a great idea to visualize the graph. If you were to graph y = (x - 5)(x - 3)(x - 4), you would see that the curve crosses the x-axis at x = 3, x = 4, and x = 5. Visualizing the graph provides a powerful confirmation of our algebraic solutions. It allows us to see the x-axis intercepts as tangible points on a curve, connecting the abstract algebra to a concrete visual representation. This visual connection deepens our understanding of the relationship between the equation and its graph, making the concept more intuitive and memorable. Graphing the equation can also help identify potential errors in our calculations, as any deviation from the expected intercepts would indicate a mistake in the algebraic process. By integrating visualization into our problem-solving approach, we create a more holistic and comprehensive understanding of the mathematical concepts involved.

Key Takeaways

Let's recap the key takeaways from this problem:

  1. X-axis intercepts are the points where the graph crosses the x-axis (where y = 0).
  2. To find x-axis intercepts, factor the equation.
  3. Set each factor equal to zero and solve for x.
  4. The solutions are the x-coordinates of the intercepts.
  5. Visualize the graph to confirm your solutions.

Finding x-axis intercepts is a fundamental skill in algebra. By mastering these steps, you'll be well-equipped to tackle more complex problems. Remember, practice makes perfect! So, try working through similar problems to build your confidence and solidify your understanding. These key takeaways serve as a roadmap for solving similar problems in the future. They highlight the essential steps and concepts that are crucial for finding x-axis intercepts effectively. By internalizing these points, we can approach new problems with a clear strategy and a solid foundation of knowledge. Regular review of these takeaways will reinforce our understanding and ensure that we are well-prepared to apply these skills in various mathematical contexts. These key takeaways are not just a summary of the solution; they are a valuable tool for future learning and problem-solving.

Practice Problems

Want to test your understanding? Try finding the x-axis intercepts of these equations:

  1. y = (x + 2)(x - 1)(x - 3)
  2. y = x(x^2 - 4x + 3)
  3. y = (x - 6)(x^2 - 5x - 6)

These practice problems offer an opportunity to apply the concepts and techniques we've learned in this guide. By working through these exercises, you can reinforce your understanding and identify any areas where you may need further clarification. Practice is essential for mastering any mathematical skill, and these problems provide a valuable opportunity to hone your abilities in finding x-axis intercepts. Don't hesitate to revisit the steps and explanations in this guide as you work through these problems. Remember, the key is to practice consistently and to approach each problem with a clear and methodical strategy. By tackling these practice problems, you'll build confidence and develop a deeper understanding of the concepts involved.

I hope this guide has helped you understand how to find x-axis intercepts. Keep practicing, and you'll become a pro in no time! Good luck, and happy problem-solving! Remember, math can be fun, especially when you break it down step by step. Keep exploring, keep learning, and keep challenging yourself. You've got this! This concluding section serves as a final encouragement and motivation for continued learning. It reinforces the idea that practice and persistence are key to mastering mathematical concepts. By expressing enthusiasm and support, we aim to inspire readers to continue their mathematical journey with confidence and a positive attitude. The final message emphasizes the enjoyable aspects of problem-solving and encourages a growth mindset, highlighting the importance of continuous exploration and learning in the field of mathematics.