Solving For X: When Function Sums Equal Zero

Hey guys! Let's dive into a cool math problem where we figure out the value of x when the sum of two functions equals zero. Sounds fun, right? We're going to break down how to do this step-by-step, making it super easy to understand. This is a common type of problem you might encounter in algebra, so mastering it will be a big win for your math skills. We'll start with our two functions, f(x) and g(x), and then we'll walk through the process of combining them, setting the result to zero, and solving for x. Get ready to flex those math muscles! This concept is fundamental in understanding how functions interact and is a building block for more advanced mathematical topics. By the end of this, you'll be able to confidently tackle similar problems. So, let's get started and make math a little less intimidating, shall we?

Understanding the Functions: f(x) and g(x)

Alright, first things first, let's get familiar with our two functions. We have f(x) = x² - 2x and g(x) = 6x + 4. Think of these functions as little machines. When you put a value of x into the f(x) machine, it squares x, subtracts two times x, and spits out a result. The g(x) machine takes x, multiplies it by six, adds four, and gives you a different result. The key here is understanding what these functions actually do. It's like having two different recipes. Each function transforms the input (x) into a new output. The shape of the graph of f(x) is a parabola and the graph of g(x) is a straight line. Visualizing these functions on a graph can sometimes help, but for this problem, we're focusing on the algebraic side. Before we move on, make sure you understand the individual function operations, as this forms the basis for the next steps. Make sure to understand the different operations that are happening in each function. Because you need to know how they work independently so you can understand what to do when they are combined. Getting a solid grasp here will make the rest of the problem so much easier. So, take a moment to review the functions and make sure you're comfortable with how they operate. This foundational knowledge is crucial before we combine them.

Now, let's go over some of the most basic principles. In mathematics, a function is a relationship or expression involving one or more variables. Usually, functions are represented by symbols, such as f(x), g(x), etc. A function has a domain, a range, and a rule that maps each element of the domain to a unique element of the range. The domain represents all the possible inputs to the function, while the range represents all the possible outputs. For example, in our problem, both functions are defined for all real numbers. This means that any real number can be plugged into f(x) or g(x). The output depends on the value of x. The functions we are going to use are quadratic and linear. f(x) is a quadratic function, and its graph is a parabola. g(x) is a linear function, and its graph is a straight line. The rules for each function are different, but the goal is the same: to find out the output for any given input, x.

Combining the Functions: (f + g)(x)

Okay, now the fun part! We need to combine these two functions. The expression (f + g)(x) means we're going to add the outputs of f(x) and g(x) together. So, we simply add the two functions. You can rewrite (f + g)(x) as f(x) + g(x). This means that for every value of x, we are going to add the value of f(x) to the value of g(x). Pretty straightforward, right?

Let's get down to the nitty-gritty. We already know that f(x) = x² - 2x and g(x) = 6x + 4. So, (f + g)(x) = (x² - 2x) + (6x + 4). See? We're just putting them together. Now we need to simplify this expression by combining like terms. Like terms are those that have the same variable raised to the same power. In our case, the like terms are the x terms and the constants. Let's combine them: -2x and 6x become 4x. The constant terms are 4, which stays the same. So, our simplified expression becomes x² + 4x + 4. This is the combined function (f + g)(x). Good job, guys! Understanding how to add and simplify functions is really important for solving more complex problems. Remember, we're not changing the functions, just combining them into a single, simplified expression. This is our new combined function, and we'll use it to find the x value where the sum is equal to zero.

Setting the Combined Function to Zero: (f + g)(x) = 0

Alright, we're getting closer to the finish line! The problem asks us to find the value of x where (f + g)(x) = 0. We now know that (f + g)(x) = x² + 4x + 4. So, we can set that expression equal to zero: x² + 4x + 4 = 0. This is a quadratic equation, which means we're going to solve for x using different methods. There are multiple ways to solve a quadratic equation. We can factor, complete the square, or use the quadratic formula. Let's try factoring first because it's usually the easiest method if the equation is factorable. We need to find two numbers that multiply to 4 (the constant term) and add up to 4 (the coefficient of the x term). If you think about it, the numbers 2 and 2 fit the bill (2 * 2 = 4 and 2 + 2 = 4). So, we can factor the quadratic equation. So we have that x² + 4x + 4 can be factored into (x + 2)(x + 2) = 0.

Great, we've successfully factored our quadratic equation! Now, it's time to solve for x. When the product of two factors is zero, at least one of the factors must be zero. In our case, we have (x + 2)(x + 2) = 0. This means that x + 2 = 0. To solve for x, we subtract 2 from both sides of the equation, which gives us x = -2. So, the value of x that makes (f + g)(x) = 0 is -2. It's awesome when everything lines up, isn't it? Let’s recap, we combined the two functions and set the resulting expression to zero. Then we factored the quadratic equation to find the value of x.

Verifying the Solution

It's always a good idea to check your answer to make sure it's correct. We can do this by plugging x = -2 back into the original equation (f + g)(x) = 0. Remember that (f + g)(x) = x² + 4x + 4. So, we'll substitute -2 for x: (-2)² + 4(-2) + 4 = 0. Let's break it down: (-2)² = 4, 4 * -2 = -8. Thus, 4 - 8 + 4 = 0. And if you simplify it, 0 = 0. Success! This means that our solution, x = -2, is correct. Doing this check helps to avoid any errors during your calculations. You always want to verify your results to make sure that the numbers make sense in the context of the problem, so you can be sure of your work. This verification step is a crucial habit to develop in mathematics. This is one of the important parts of the math problem, and it always makes sure that you got the right answer.

Conclusion: We Did It!

Woohoo! We've successfully found the value of x where (f + g)(x) = 0. We started with two functions, combined them, set the result to zero, and solved for x. Along the way, we touched on function operations, simplifying expressions, and solving quadratic equations. This is a powerful combination of skills, and you should feel proud of your accomplishment. Keep practicing, and you'll get even better at these types of problems. Remember, math is like a muscle – the more you use it, the stronger it gets. And the more you understand these concepts, the easier future problems will be. So, keep up the great work! If you like this type of problem, be sure to find more to exercise your brain, and you can achieve your math goals, guys!