X-Intercepts Of Y=3(x-1)(x+6): A Step-by-Step Solution

by Henrik Larsen 55 views

Hey guys! Let's dive into a common algebra problem: finding the x-intercepts of a quadratic equation. Specifically, we're going to tackle the equation y = 3(x - 1)(x + 6). If you've ever felt a little confused about how to find these points, don't worry! We'll break it down into simple, easy-to-follow steps. By the end of this guide, you'll not only know the answer but also understand the why behind the process. So, grab your math tools, and let's get started!

Understanding xx-intercepts

Before we jump into solving the problem, let's make sure we're all on the same page about what xx-intercepts actually are. The x-intercepts are the points where the graph of an equation crosses the x-axis. Think of it like this: imagine you're drawing a line or a curve on a graph. The places where that line or curve touches or intersects the horizontal line (the x-axis) are the x-intercepts. These points are also sometimes called roots or zeros of the equation. Basically, they're the x-values that make the y-value equal to zero. This is a key concept, and it's what we'll use to solve our problem.

To put it mathematically, at the x-intercept, y = 0. This makes sense if you visualize the coordinate plane. Any point on the x-axis has a y-coordinate of 0. For example, the point (5, 0) is on the x-axis, and so is (-3, 0). When we're asked to find the x-intercepts, we're essentially being asked to find the x-values that make the equation equal to zero. This understanding is crucial because it tells us exactly how to approach the problem. We need to set y to 0 in our equation and then solve for x. This might sound a little abstract now, but it will become clear as we work through the specific example. So, keep in mind: x-intercepts are where y = 0, and this is our starting point for solving the problem.

Setting yy to Zero

Now that we know what x-intercepts are and why they matter, let's apply this knowledge to our equation, y = 3(x - 1)(x + 6). Remember, the first step in finding the x-intercepts is to set y equal to zero. This is because, at the x-intercept, the graph of the equation intersects the x-axis, and at any point on the x-axis, the y-coordinate is always zero. So, we're essentially finding the x-values that make the equation true when y is zero.

By setting y to zero, our equation transforms from y = 3(x - 1)(x + 6) to 0 = 3(x - 1)(x + 6). This simple substitution is a powerful move because it allows us to focus solely on the x-values that satisfy the equation. We've effectively turned the problem of finding where the graph crosses the x-axis into an algebra problem of solving for x. This is a common strategy in mathematics: transforming a problem into a form that we know how to solve. The equation 0 = 3(x - 1)(x + 6) is now in a form that we can easily work with. It tells us that the product of three factors – 3, (x - 1), and (x + 6) – must equal zero. This is where the zero-product property comes into play, which we'll discuss in the next section. But for now, the important thing is to recognize the significance of setting y to zero. It's the key that unlocks the door to finding the x-intercepts. So, always remember: to find the x-intercepts, set y to zero.

Applying the Zero-Product Property

Okay, so we've set y to zero, and our equation now looks like this: 0 = 3(x - 1)(x + 6). This is where the zero-product property comes to the rescue. The zero-product property is a fundamental concept in algebra, and it's super useful for solving equations like this one. It basically says that if the product of several factors is equal to zero, then at least one of those factors must be zero. Think about it: if you multiply a bunch of numbers together and get zero as the answer, one of those numbers had to be zero. It's a simple but powerful idea.

In our case, we have three factors: 3, (x - 1), and (x + 6). The equation 0 = 3(x - 1)(x + 6) tells us that the product of these three factors is zero. According to the zero-product property, this means that at least one of these factors must be zero. Now, 3 can never be zero, so we can ignore that factor. This leaves us with two possibilities: either (x - 1) = 0 or (x + 6) = 0. This is a crucial step because it breaks our original equation into two simpler equations that we can solve individually. We've essentially transformed one problem into two easier problems. By applying the zero-product property, we've narrowed down the possibilities and set ourselves up to find the x-intercepts. Remember, the zero-product property is your friend when dealing with factored equations like this. It's the key to unlocking the solutions.

Solving for xx

Alright, we've arrived at the final stage of our journey: solving for x. Thanks to the zero-product property, we've broken our original equation into two simpler equations: (x - 1) = 0 and (x + 6) = 0. Now, all we need to do is solve each of these equations for x. This is where our basic algebra skills come into play. Let's tackle the first equation: (x - 1) = 0. To isolate x, we simply add 1 to both sides of the equation. This gives us x = 1. So, one of our x-intercepts is x = 1.

Now, let's move on to the second equation: (x + 6) = 0. To isolate x in this equation, we subtract 6 from both sides. This gives us x = -6. So, our other x-intercept is x = -6. And that's it! We've found the two x-values that make our original equation equal to zero. These are the x-coordinates of the points where the graph of the equation y = 3(x - 1)(x + 6) crosses the x-axis. We've successfully solved for x by using basic algebraic manipulation. Remember, the goal is always to isolate the variable you're trying to find. By performing the same operation on both sides of the equation, you maintain the equality and gradually get closer to the solution. So, to recap, we found two solutions: x = 1 and x = -6. But we're not quite done yet. We need to express these solutions as coordinate points, which we'll do in the next section. Always remember the importance of isolating x to find its value.

Expressing the xx-intercepts as Coordinate Points

We've done the heavy lifting and found the x-values that make our equation equal to zero: x = 1 and x = -6. But remember, x-intercepts are points on a graph, and points are represented by coordinates (x, y). We know the x-values, but what about the y-values? Well, this is where our initial understanding of x-intercepts comes back into play. We know that at any x-intercept, the y-value is always 0. This is because the x-intercept is the point where the graph crosses the x-axis, and all points on the x-axis have a y-coordinate of 0.

So, for our first x-value, x = 1, the corresponding y-value is 0. This means one of our x-intercepts is the point (1, 0). For our second x-value, x = -6, the corresponding y-value is also 0. This means our other x-intercept is the point (-6, 0). We've now successfully expressed our solutions as coordinate points. This is the final step in finding the x-intercepts. It's important to remember to express your answers as points because that's the standard way to represent locations on a graph. The points (1, 0) and (-6, 0) tell us exactly where the graph of the equation y = 3(x - 1)(x + 6) crosses the x-axis. So, always remember to express the x-intercepts as coordinate points to give a complete and accurate answer.

The Answer

We've made it to the end, guys! After working through all the steps, we've successfully found the x-intercepts for the graph of y = 3(x - 1)(x + 6). We set y to zero, applied the zero-product property, solved for x, and expressed our solutions as coordinate points. Our final answer is that the x-intercepts are (1, 0) and (-6, 0). Looking back at the options provided, we can see that this corresponds to option D. So, the correct answer is D. (1, 0) and (-6, 0).

Let's take a moment to recap the key steps we followed: 1. Understand what x-intercepts are (where the graph crosses the x-axis, y = 0). 2. Set y to zero in the equation. 3. Apply the zero-product property to break the equation into simpler equations. 4. Solve for x in each equation. 5. Express the solutions as coordinate points (x, 0). By following these steps, you can confidently find the x-intercepts of any equation in this form. Remember, practice makes perfect! The more you work through problems like this, the more comfortable and confident you'll become. So, keep practicing, and you'll be a pro at finding x-intercepts in no time!