Multiplying Expressions: A Step-by-Step Guide

Hey everyone! Today, we're diving into a common math problem: multiplying expressions. Specifically, we're going to solve $(3-\sqrt{5})(2+\sqrt{5})$. Don't worry if this looks a bit intimidating at first; we'll break it down into easy-to-follow steps. This type of problem is super important for understanding algebra and working with radicals (the fancy name for square roots). Let's get started, shall we?

Understanding the Problem: The Basics of Multiplication

First things first, let's make sure we're all on the same page. The expression $(3-\sqrt{5})(2+\sqrt{5})$ is asking us to multiply two different sets of numbers. Each set has a regular number and a square root. Remember, when you see something like this, you need to apply the distributive property, sometimes also known as the FOIL method, which stands for First, Outer, Inner, Last. This helps us ensure we multiply everything correctly. It's like making sure you shake everyone's hand at a party – you don't want to miss anyone! The distributive property is your best friend here. It's the key to making sure every term in the first set is multiplied by every term in the second set. So, we'll multiply 3 by 2, 3 by $\sqrt{5}$, $-\sqrt{5}$ by 2, and $-\sqrt{5}$ by $\sqrt{5}$. Easy peasy, right?

Before we jump into the calculation, let’s quickly recap why understanding the distributive property is crucial. It's not just about solving this particular problem; it’s about building a solid foundation in algebra. The distributive property is used in countless scenarios, from simplifying equations to expanding expressions and even solving more complex problems later on. Mastering this now will make your future math endeavors significantly smoother. Now, let’s get into the step-by-step process of solving this expression. By the end of this, you’ll be handling these types of problems like a pro! So, are you ready to become a multiplication master? Let’s do it!

Step-by-Step Solution: Breaking Down the Multiplication

Alright, let's break this down step-by-step. We'll start with the FOIL method to make sure we don't miss anything. We're going to multiply $(3-\sqrt{5})(2+\sqrt{5})$:

1. First: Multiply the first terms in each set: $3 * 2 = 6$
2. Outer: Multiply the outer terms: $3 * \sqrt{5} = 3\sqrt{5}$
3. Inner: Multiply the inner terms: $-\sqrt{5} * 2 = -2\sqrt{5}$
4. Last: Multiply the last terms in each set: $-\sqrt{5} * \sqrt{5} = -5$

Now, let's put it all together. We have $6 + 3\sqrt{5} - 2\sqrt{5} - 5$. Notice how we've carefully multiplied each term to ensure we account for all the values involved. This is where the magic happens. We've taken a seemingly complex expression and broken it down into smaller, manageable pieces.

Then, we combine like terms. Like terms are those that have the same variable or, in this case, the same radical. We can combine the whole numbers (6 and -5) and the terms with $\sqrt{5}$ ($\sqrt{5}$ and $-2\sqrt{5}$). We will put all of our calculations together. We're going to bring them together to create a simplified answer. Let’s do it now!

Simplifying the Expression: Combining Like Terms

Now that we've done the multiplication, it's time to simplify. We need to combine the like terms to get our final answer. Remember, like terms are terms that have the same variable or radical. In our expression, $6 + 3\sqrt{5} - 2\sqrt{5} - 5$, we have:

• Whole numbers: 6 and -5
• Terms with $\sqrt{5}$: $3\sqrt{5}$ and $-2\sqrt{5}$

Let's combine these:

• Combine the whole numbers: $6 - 5 = 1$
• Combine the terms with $\sqrt{5}$: $3\sqrt{5} - 2\sqrt{5} = 1\sqrt{5}$ or simply $\sqrt{5}$

So, when we combine everything, we get $1 + \sqrt{5}$. This is the simplified form of our original expression. Isn't that neat? It shows how we went from a slightly complex starting point to a concise and simplified final answer. Combining like terms is a core skill in algebra, which helps make equations easier to understand and work with. So, take pride in knowing that you've not only solved the problem, but also strengthened a vital mathematical skill.

Now, let's celebrate. You have successfully simplified the expression $(3-\sqrt{5})(2+\sqrt{5})$ to $1 + \sqrt{5}$. That's the final answer! Pat yourself on the back, you’ve earned it!

Conclusion: The Final Answer and Key Takeaways

So, what's the answer? The simplified form of $(3-\sqrt{5})(2+\sqrt{5})$ is $1 + \sqrt{5}$. Congratulations, guys! You've successfully navigated the multiplication of expressions involving square roots. This problem is a great example of how to break down complex problems into manageable steps using the distributive property and combining like terms. You've also gained more confidence in your algebra skills. Great job! The key takeaways are:

• Distributive Property (FOIL): Always remember to multiply each term in the first set by each term in the second set.
• Combining Like Terms: Simplify your expression by combining terms that have the same radical or no radical.
• Practice Makes Perfect: Keep practicing these types of problems to become more comfortable and efficient. The more you practice, the easier it becomes.

And there you have it! You've not only solved the problem but also learned and reinforced key mathematical concepts. Multiplication problems with radicals are super fun once you get the hang of it, and this understanding will be useful for many more complex math problems that you'll come across. Keep up the awesome work, and keep practicing. I am certain that with regular practice, you’ll find that algebra and radical simplification become second nature. Until next time, keep crunching those numbers!