Calculate 3a² − 2b + 22 With A = 2 And B = -3

Hey guys! Today, we're diving into a fun math problem. We're going to calculate the value of the algebraic expression 3a² − 2b + 22 when a equals 2 and b equals -3. Don't worry, it's not as scary as it looks! We'll break it down step by step so it's super easy to follow. Understanding algebraic expressions is super important, it is a fundamental concept in mathematics that bridges the gap between basic arithmetic and more advanced topics like calculus and linear algebra. Mastering these concepts opens doors to various fields, including engineering, computer science, economics, and data analysis. When you grasp how to manipulate and evaluate expressions, you're not just solving equations; you're developing a powerful problem-solving toolkit. Algebraic expressions are like the building blocks of mathematical language. Think of them as sentences where numbers and symbols come together to represent relationships and quantities. In an expression, you'll find variables (like a and b in our case) that stand for unknown values, constants (numbers that don't change), and operations (like addition, subtraction, multiplication, and division). Learning to work with these expressions means you can describe real-world situations mathematically, predict outcomes, and make informed decisions. Imagine you're trying to figure out the total cost of a project, plan a budget, or model the trajectory of a rocket – algebraic expressions are your best friends. In this article, we will explore the step-by-step process of substituting values into expressions, simplifying them using the order of operations, and arriving at the final result. So, let’s get started and tackle this algebraic expression together!

Understanding the Expression

Before we jump into the calculation, let's really understand what the expression 3a² − 2b + 22 is telling us. This expression has three main parts, or terms, each involving different operations. The first term is 3a², which means 3 multiplied by a squared (a multiplied by itself). The second term is -2b, which means -2 multiplied by b. And the last term is +22, which is a constant—it's just a plain old number. Breaking down the expression like this helps us see the individual components we need to deal with. Think of it like looking at the blueprint of a building before you start construction. You need to understand each part before you can put it all together. In mathematics, this is crucial because the order in which you perform operations matters. You can't just do things in any order you like; there's a specific set of rules we follow to ensure we get the correct answer. These rules are known as the order of operations, and they're the key to simplifying algebraic expressions effectively. Now, let's talk about each component in a bit more detail. The term 3a² involves an exponent (the little 2 above the a), which means we need to square a first before multiplying by 3. This is where the order of operations comes into play, and we'll see exactly how it works in the next section. The term -2b is a straightforward multiplication. We're simply multiplying -2 by the value of b. But remember, the negative sign is important! It changes the sign of the result, so we need to keep track of it carefully. And finally, the constant term +22 is just a fixed value. It doesn't depend on a or b, so we can simply add it to the result after we've calculated the other terms. By understanding these individual components and how they interact, we're setting ourselves up for success in simplifying the entire expression. Now, let's move on to the next step: substituting the given values for a and b. This is where we replace the variables with the numbers we've been given, and the expression starts to take on a concrete numerical value. So, let's get to it!

Substituting the Values

Okay, so we know that a = 2 and b = -3. The next step is to substitute these values into our expression, 3a² − 2b + 22. This means we replace every instance of a with 2 and every instance of b with -3. When you substitute, it's a good idea to use parentheses, especially with negative numbers. This helps to keep the signs clear and avoid mistakes. So, when we substitute, our expression becomes 3(2)² − 2(-3) + 22. See how we've replaced a with (2) and b with (-3)? The parentheses are crucial here. They tell us that the numbers inside are being multiplied. Without the parentheses, it's easy to misinterpret the expression, especially when there are negative signs involved. Substitution is a fundamental skill in algebra, it allows us to take an abstract expression and turn it into something concrete that we can evaluate. It's like having a recipe with variable ingredients – you can't bake the cake until you know the exact amounts of each ingredient. In this case, a and b are our variable ingredients, and 2 and -3 are their specific amounts. Once we substitute these values, we're ready to start cooking! Now, let's talk a bit more about why parentheses are so important. They're not just there for show; they serve a vital purpose in mathematics. Parentheses tell us to perform the operations inside them first. This is a key part of the order of operations, which we'll discuss in more detail in the next section. But for now, just remember that parentheses group things together, and we need to deal with them before anything else. In our expression, 3(2)² means that we need to square 2 first, and then multiply the result by 3. Similarly, -2(-3) means that we need to multiply -2 by -3. Without the parentheses, we might end up doing the operations in the wrong order, and we'd get the wrong answer. Another important thing to remember when substituting is to be careful with the signs. Negative signs can be tricky, and it's easy to make a mistake if you're not paying attention. That's why using parentheses is such a good habit – it helps to keep the signs organized and reduces the risk of errors. So, now that we've substituted the values of a and b into our expression, we're ready to move on to the next step: simplifying the expression. This is where we use the order of operations to perform the calculations and arrive at our final answer. Let's see how it's done!

Applying the Order of Operations (PEMDAS/BODMAS)

Alright, we've substituted the values, and now we have 3(2)² − 2(-3) + 22. To simplify this, we need to follow the order of operations, which you might remember as PEMDAS or BODMAS. Both acronyms stand for the same thing: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is crucial, it ensures that we perform the operations in the correct sequence, leading to the accurate result. Think of PEMDAS/BODMAS as the rules of the road for mathematical calculations. Just like you need to follow traffic laws to drive safely, you need to follow the order of operations to simplify expressions correctly. If you skip a step or do things out of order, you're likely to end up at the wrong destination – in this case, the wrong answer. So, let's break down our expression using PEMDAS/BODMAS. First up, we have Parentheses/Brackets. In our expression, we have parentheses around the numbers 2 and -3, but there are no operations to perform inside them. So, we can move on to the next step. Next, we have Exponents/Orders. We have one exponent in our expression: 2², which means 2 squared, or 2 multiplied by itself. So, we calculate 2² = 2 * 2 = 4. Now our expression looks like this: 3(4) − 2(-3) + 22. See how we've simplified the exponent? We're making progress! Now, we move on to Multiplication and Division. We have two multiplications in our expression: 3(4) and -2(-3). Let's do them from left to right. First, 3(4) = 3 * 4 = 12. Then, -2(-3) = -2 * -3 = 6. Remember that a negative number multiplied by a negative number gives a positive result. Now our expression looks like this: 12 + 6 + 22. We're almost there! Finally, we have Addition and Subtraction. We only have addition in our expression, so we can simply add the numbers together from left to right. First, 12 + 6 = 18. Then, 18 + 22 = 40. And that's it! We've simplified the expression using the order of operations, and we've arrived at our final answer: 40. The order of operations is more than just a set of rules, it's a fundamental principle of mathematics. It ensures consistency and clarity in calculations, and it's essential for solving complex problems. By mastering PEMDAS/BODMAS, you're building a solid foundation for your mathematical journey. So, now that we've successfully applied the order of operations to our expression, let's take a moment to summarize our steps and celebrate our achievement!

The Final Result

Okay, guys, we've done it! We've successfully calculated the value of the algebraic expression 3a² − 2b + 22 when a = 2 and b = -3. We followed all the steps carefully: we understood the expression, we substituted the values, we applied the order of operations, and now we have our final result. Drumroll, please… The value of the expression is 40. Woohoo! Isn't it satisfying when you solve a problem like this? You started with a seemingly complex expression, but by breaking it down into smaller steps and following the rules, you arrived at a clear and concise answer. This is the power of algebra! It allows us to take abstract ideas and turn them into concrete solutions. Now, let's recap the steps we took to get here. First, we understood the expression 3a² − 2b + 22. We identified the different terms and operations involved, and we saw how they all fit together. This is like understanding the ingredients in a recipe before you start cooking. Next, we substituted the values of a and b. We replaced a with 2 and b with -3, using parentheses to keep the signs clear. This is like adding the specific amounts of each ingredient to the mixing bowl. Then, we applied the order of operations (PEMDAS/BODMAS). We started with the exponent, then moved on to multiplication, and finally addition. This is like following the cooking instructions step by step, making sure to do everything in the right order. And finally, we arrived at our final result: 40. This is like taking the finished cake out of the oven – the culmination of all our hard work! Understanding algebraic expressions and how to evaluate them is a crucial skill in mathematics. It's the foundation for more advanced topics, and it's also incredibly useful in real-world situations. Whether you're calculating the cost of something, planning a budget, or modeling a scientific phenomenon, algebra can help you make sense of the world around you. So, congratulations on mastering this problem! You've taken a big step in your mathematical journey. And remember, the key to success in math is to break down complex problems into smaller, manageable steps, and to follow the rules and principles you've learned. Now that we've conquered this expression, let's keep exploring the fascinating world of algebra and see what other challenges we can overcome!

Why This Matters

You might be wondering,