Unveiling The Coefficient: Simplifying (-x-5)^2

Hey math enthusiasts! Let's dive into a fun problem: figuring out the coefficient of x after we simplify the expression (-x - 5)^2. Don't worry, it's not as scary as it looks! We'll break it down step by step, making sure everyone understands the process. So, grab your pens and paper, and let's get started. This question is a classic example of algebra, and it tests your understanding of expanding and simplifying expressions. Mastering this skill is super important as you journey through more complex math problems down the line. Ready to learn how to find the coefficient of x? Awesome, let's go!

Understanding the Basics: What's a Coefficient Anyway?

Before we jump into the problem, let's make sure we're all on the same page. What exactly is a coefficient? Simply put, the coefficient is the number that multiplies a variable in an algebraic expression. For example, in the expression 3x, the coefficient of x is 3. In 7y, the coefficient of y is 7. Easy peasy, right? Coefficients can be positive, negative, whole numbers, fractions, or even decimals. They are the numerical factors that tell us how many of each variable we have. So, when we talk about finding the coefficient of x in our problem, we are looking for the number that will be directly in front of the x after we simplify the expression. Got it? Cool! Now, let's move on to the expression (-x - 5)^2.

Expanding the Expression: Step-by-Step Guide

Now, let's expand the expression (-x - 5)^2. This means we need to multiply the expression by itself. Remember that squaring something means multiplying it by itself. So, (-x - 5)^2 is the same as (-x - 5) * (-x - 5). Now, let's use the distributive property (also known as the FOIL method – First, Outer, Inner, Last) to multiply these two binomials. Don't worry; it's not as complicated as it sounds. First, we multiply the First terms: -x * -x = x^2. Then, we multiply the Outer terms: -x * -5 = 5x. Next, we multiply the Inner terms: -5 * -x = 5x. Finally, we multiply the Last terms: -5 * -5 = 25. So, putting it all together, we get x^2 + 5x + 5x + 25. See? Not so bad, right? We've successfully expanded the expression!

Simplifying the Expression: Combining Like Terms

Now that we've expanded the expression, the next step is to simplify it. Simplifying means combining like terms. In our expanded expression, x^2 + 5x + 5x + 25, the like terms are 5x and 5x. Combining them, we get 5x + 5x = 10x. So, our simplified expression becomes x^2 + 10x + 25. Notice how we've reduced the number of terms and made the expression easier to work with. Simplifying expressions is a fundamental skill in algebra, making complex problems manageable. This will make the equation easier to read and use for different mathematical purposes. And now, we're almost at the finish line!

Identifying the Coefficient of x: The Grand Finale

Finally, we're at the heart of the problem: finding the coefficient of x in our simplified expression, which is x^2 + 10x + 25. Remember, the coefficient is the number that multiplies the variable. In the term 10x, the coefficient is 10. That's because 10 is directly in front of the x. So, the coefficient of x in the expression x^2 + 10x + 25 is 10. And there you have it! We've successfully expanded, simplified, and found the coefficient of x. Great job, everyone! This is an important step in learning how to manipulate and understand algebraic expressions.

Why This Matters: Real-World Applications

You might be wondering, "Why does this even matter?" Well, understanding and manipulating algebraic expressions like these has many real-world applications. It's used in fields like engineering, physics, computer science, and even economics. For instance, engineers use algebraic expressions to calculate the forces acting on a bridge, physicists use them to model the motion of objects, and computer scientists use them in programming to create algorithms. Even in everyday life, understanding these concepts can help you analyze data, make informed decisions, and solve problems more effectively. So, while it might seem like just a math problem, the skills you learn here can be applied in countless ways. It also aids in higher-level math, such as calculus.

Quick Recap and Conclusion

Alright, let's quickly recap what we did. First, we understood what a coefficient is. Then, we expanded the expression (-x - 5)^2 to get x^2 + 5x + 5x + 25. Next, we simplified the expression by combining like terms, resulting in x^2 + 10x + 25. Finally, we identified the coefficient of x, which is 10. See? It's a simple process, but it's a fundamental skill in algebra. Keep practicing, and you'll become a pro in no time! Remember that mathematics is all about building on your skills, so each concept you learn will help you prepare for the next, more complex ideas that are to come. Thanks for joining me on this mathematical journey! Keep practicing, and have fun with it. Keep exploring, and you'll be amazed at how many fascinating mathematical concepts are out there. Until next time, happy calculating!

Answer

So, the correct answer is C. 10