X-Intercepts: Find Where Graph Crosses X-Axis

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Hey guys! Let's figure out where the graph of the equation y=(x-5)(x^2-7x+12) crosses the x-axis. These points are super important in understanding the behavior of the graph, and they're called the x-intercepts or roots of the equation. Basically, we’re looking for the values of x that make y equal to zero. This is where the magic happens, and the graph intersects the x-axis. So, let’s dive in and break down this problem step by step, making it super clear and easy to understand.

Setting y to Zero

First things first, to find where the graph crosses the x-axis, we need to set y to zero. Why? Because any point on the x-axis has a y-coordinate of zero. So, our equation becomes:

0 = (x-5)(x^2-7x+12)

Now, this looks a bit intimidating, but don't worry, we're going to simplify it. We have a product of two factors here: (x-5) and (x^2-7x+12). If either of these factors equals zero, then the whole equation equals zero. This is a key concept that will help us solve the problem. Think of it like this: if you have two numbers, and their product is zero, then at least one of those numbers must be zero. This simple idea allows us to break down the problem into smaller, more manageable parts.

The first factor is straightforward: (x-5). If x-5 = 0, then we have one of our solutions. The second factor, (x^2-7x+12), is a quadratic equation. We'll need to factor this to find its roots. Factoring a quadratic equation might seem tricky at first, but with a little practice, it becomes second nature. We’re looking for two numbers that multiply to 12 and add up to -7. These numbers will help us break down the quadratic into two binomial factors. Once we have these factors, we can set each one to zero and solve for x, giving us the remaining solutions. So, let's move on to factoring the quadratic and uncovering the rest of the x-intercepts.

Factoring the Quadratic Equation

Okay, let's tackle the quadratic part of our equation: x^2-7x+12. To factor this, we need to find two numbers that multiply to 12 (the constant term) and add up to -7 (the coefficient of the x term). Take a moment to think about the factors of 12: 1 and 12, 2 and 6, 3 and 4. Which pair, when adjusted with negative signs, will add up to -7? You got it – it’s -3 and -4!

So, we can rewrite the quadratic equation as:

(x-3)(x-4)

This is the factored form of our quadratic equation. Now, we have the entire original equation in a factored form:

0 = (x-5)(x-3)(x-4)

This is awesome! We’ve broken down the problem into three simple factors. Remember, if the product of these factors is zero, then at least one of them must be zero. This means we can set each factor equal to zero and solve for x. By finding the values of x that make each factor zero, we’ll find all the points where the graph crosses the x-axis. This step is crucial because it transforms the problem from a complex equation into a set of simple linear equations, each of which is easy to solve. So, let's move on to the final step: solving for x in each of these factors and discovering our x-intercepts.

Solving for x

Now comes the fun part – solving for x! We have three factors: (x-5), (x-3), and (x-4). Let’s set each of these equal to zero and solve:

  1. x - 5 = 0

Adding 5 to both sides, we get:

x = 5

  1. x - 3 = 0

Adding 3 to both sides, we get:

x = 3

  1. x - 4 = 0

Adding 4 to both sides, we get:

x = 4

So, we have three values for x: 5, 3, and 4. These are the x-coordinates of the points where the graph crosses the x-axis. To represent these as points, we pair each x-value with a y-value of 0 (since these points are on the x-axis). This gives us the points (5, 0), (3, 0), and (4, 0). These points are the x-intercepts of the graph. They tell us exactly where the graph of the equation intersects the x-axis. Understanding these intercepts is crucial for sketching the graph and understanding the function's behavior. So, we’ve successfully found the points where the graph crosses the x-axis. Let's summarize our findings and wrap up the problem.

Conclusion

Alright guys, we've successfully found the points where the graph of y=(x-5)(x^2-7x+12) crosses the x-axis! By setting y to zero, factoring the equation, and solving for x, we discovered the x-intercepts.

The x-intercepts are:

  • (5, 0)
  • (3, 0)
  • (4, 0)

These points are where the graph intersects the x-axis, giving us valuable information about the function's behavior. Understanding how to find x-intercepts is a fundamental skill in algebra and calculus. It helps us visualize the graph of a function and understand its properties. We broke down a potentially complex problem into manageable steps, making it clear and straightforward. Remember, the key is to set y to zero, factor the equation, and solve for x. With a little practice, you’ll be finding x-intercepts like a pro! Keep up the great work, and let's tackle more math problems together! Finding these intercepts not only helps in mathematics but also in real-world applications where graphs are used to model various scenarios. So, mastering this skill is super beneficial. Let’s keep exploring and making math fun and accessible for everyone!