Solving (f+g)(x) = 0 When F(x) = X^2 - 2x And G(x) = 6x + 4

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Hey there, math enthusiasts! Today, we're diving into a fun little problem involving function composition and finding roots. We've got two functions, f(x) = x² - 2x and g(x) = 6x + 4, and our mission, should we choose to accept it, is to find the value(s) of x for which (f + g)(x) = 0. Sounds like a plan? Let's get started!

Understanding the Problem: Deconstructing (f + g)(x)

Before we jump into calculations, let's break down what (f + g)(x) actually means. Simply put, it's the sum of the two functions f(x) and g(x). So, we can write it as:

(f + g)(x) = f(x) + g(x)

This means we're going to add the expressions for f(x) and g(x) together. Once we've done that, we'll have a new expression, and our goal is to find the x value(s) that make this new expression equal to zero. These values are also known as the roots or zeros of the function. Finding roots is a fundamental concept in algebra and calculus, and it has applications in various fields, from physics and engineering to economics and computer science. When we set a function equal to zero, we're essentially asking, "Where does this function cross the x-axis?" The points where the function intersects the x-axis are the roots of the equation. In practical terms, finding roots can help us solve optimization problems, model physical systems, and make predictions based on mathematical models. For example, in physics, finding the roots of an equation might tell us when a projectile hits the ground or when a circuit reaches a stable state. In economics, roots can represent equilibrium points in supply and demand curves. And in computer science, roots can be used to find solutions to complex algorithms and equations. Understanding how to find roots is therefore a crucial skill in many areas of study and work. Now that we've got a solid grasp of the concept, let's move on to the nitty-gritty of solving our specific problem.

Step-by-Step Solution: Putting the Pieces Together

1. Find (f + g)(x)

Alright, let's add those functions together. We know:

  • f(x) = x² - 2x
  • g(x) = 6x + 4

So,

(f + g)(x) = (x² - 2x) + (6x + 4)

Now, we simplify by combining like terms:

(f + g)(x) = x² - 2x + 6x + 4 (f + g)(x) = x² + 4x + 4

Great! We've got our combined function. This resulting quadratic expression is much simpler to work with, and it sets the stage for us to find the values of x that make the whole thing equal to zero. The process of combining like terms is a fundamental algebraic skill that allows us to simplify complex expressions and make them easier to manipulate. In this case, by combining the x terms (-2x and 6x), we've reduced the expression to a standard quadratic form, which we can then solve using a variety of methods, such as factoring, completing the square, or using the quadratic formula. Each of these methods has its own advantages and disadvantages, depending on the specific quadratic equation we're dealing with. For example, factoring is often the quickest method if the quadratic expression can be easily factored, while the quadratic formula is a more general method that works for any quadratic equation. Understanding these different methods and when to apply them is a crucial part of mastering algebra. Now that we have our simplified expression for (f + g)(x), let's move on to the next step: setting it equal to zero and solving for x.

2. Set (f + g)(x) = 0

Now for the crucial step: setting our combined function equal to zero:

x² + 4x + 4 = 0

We now have a quadratic equation. There are several ways to solve these, but let's try factoring first. Factoring involves rewriting the quadratic expression as a product of two binomials. If we can find two binomials that multiply together to give us x² + 4x + 4, then we can easily find the roots of the equation by setting each binomial equal to zero. Factoring is a powerful technique because it allows us to break down a complex equation into simpler parts, making it easier to solve. However, not all quadratic equations can be easily factored. In those cases, we can turn to other methods, such as completing the square or using the quadratic formula. Completing the square involves manipulating the quadratic equation to create a perfect square trinomial, which can then be easily factored. The quadratic formula, on the other hand, is a general formula that gives us the solutions to any quadratic equation, regardless of whether it can be factored or not. This formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0, and it's a fundamental tool in algebra. Understanding these different methods for solving quadratic equations is essential for success in mathematics and many related fields. In our case, however, we're in luck: the quadratic expression x² + 4x + 4 is indeed factorable. Let's see how we can do it.

3. Solve for x

Can you see how the left side factors? It's a perfect square trinomial!

(x + 2)(x + 2) = 0

Or, we can write it as:

(x + 2)² = 0

To find the value of x, we set the factor equal to zero:

x + 2 = 0

Subtract 2 from both sides:

x = -2

And there we have it! The value of x that makes (f + g)(x) = 0 is x = -2. This means that the graph of the quadratic function (f + g)(x) = x² + 4x + 4 touches the x-axis at exactly one point, x = -2. In other words, the function has a double root at x = -2. Understanding the concept of double roots is important in calculus and other advanced mathematical topics. A double root indicates that the function has a special kind of behavior at that point – it touches the x-axis but doesn't cross it. This can have implications for the function's derivatives and its overall shape. Now, let's take a moment to recap what we've done and make sure we've answered the question fully.

Conclusion: The Answer and the Journey

So, guys, we've successfully navigated this problem! We found that when f(x) = x² - 2x and g(x) = 6x + 4, (f + g)(x) = 0 when x = -2. Remember, the key was to first find the expression for (f + g)(x) by adding the two functions, and then solving the resulting quadratic equation. This problem illustrates the importance of understanding function composition and being comfortable with algebraic manipulation. It also highlights the power of factoring as a method for solving quadratic equations. But remember, factoring isn't always the easiest or most efficient method. Sometimes, completing the square or using the quadratic formula is a better approach. The more tools you have in your mathematical toolkit, the better equipped you'll be to tackle a wide range of problems. And don't forget, practice makes perfect! The more you work through problems like this, the more confident and comfortable you'll become with these concepts. Now, armed with this knowledge, you're ready to take on even more challenging mathematical adventures. Keep exploring, keep learning, and most importantly, keep having fun with math! Whether you're solving equations, graphing functions, or delving into more advanced topics like calculus and differential equations, the skills you develop in algebra will serve you well. So keep practicing, keep challenging yourself, and never stop learning. Math is a journey, not a destination, and there's always something new to discover. And who knows? Maybe the next problem you solve will unlock a whole new level of understanding and open doors to exciting opportunities in your future. So go out there and make some mathematical magic happen!