Calculate (a + B)(a - B) If A² = 5, B² = 6

Hey guys! Let's dive into this math problem where we need to calculate the value of the expression (a + b)(a - b), given that a² = 5 and b² = 6. It might seem a bit tricky at first, but trust me, we'll break it down step by step and you'll see it's actually pretty straightforward. We'll also explore why this type of problem is important and where you might encounter it in real life. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we fully understand what the problem is asking. We are given two equations: a² = 5 and b² = 6. Our main goal is to find the value of the expression (a + b)(a - b). This expression looks familiar, right? It's actually a special algebraic identity, which we'll discuss in more detail later. The problem also presents us with multiple-choice options (a) 1, (b) 2, (c) 3, and (d) 4, so we know we're looking for a numerical answer. This is a classic algebra problem that tests our understanding of algebraic identities and how to manipulate equations. You might encounter similar problems in algebra classes, standardized tests, or even in practical situations where you need to simplify expressions. The key here is to recognize the structure of the expression and use the given information effectively.

Step 1: Recognizing the Algebraic Identity

The most crucial step in solving this problem is recognizing the algebraic identity in the expression (a + b)(a - b). This is a classic difference of squares identity. The difference of squares identity states that (x + y)(x - y) = x² - y². It's a fundamental concept in algebra and is used extensively in simplifying expressions and solving equations. In our case, x corresponds to 'a' and y corresponds to 'b'. So, (a + b)(a - b) is equivalent to a² - b². Recognizing this identity significantly simplifies our problem because instead of having to find the individual values of 'a' and 'b' and then perform the multiplication, we can directly substitute the given values of a² and b² into the simplified expression. This not only saves us time but also reduces the chances of making errors in calculations. Understanding and recognizing common algebraic identities like this one is a valuable skill in mathematics. They appear frequently in various contexts, from basic algebra to more advanced topics like calculus. So, mastering these identities is definitely worth the effort!

Step 2: Applying the Identity and Substituting Values

Now that we've recognized the difference of squares identity, let's apply it to our problem. We know that (a + b)(a - b) = a² - b². This is where the given information comes in handy. We are given that a² = 5 and b² = 6. All we need to do is substitute these values into the simplified expression. Replacing a² with 5 and b² with 6, we get: a² - b² = 5 - 6. This is a straightforward subtraction problem. The substitution step is crucial because it transforms a seemingly complex expression into a simple arithmetic problem. It highlights the power of algebraic identities in simplifying mathematical expressions. By recognizing the pattern and applying the appropriate identity, we avoid the need to calculate the square roots of 5 and 6 (which would involve irrational numbers) and then perform the multiplications. This approach is much more efficient and less prone to errors. So, always be on the lookout for opportunities to use algebraic identities – they can be your best friend in math!

Step 3: Calculating the Final Result

We've simplified the expression to 5 - 6, so now it's time to do the final calculation. Subtracting 6 from 5 gives us: 5 - 6 = -1. Therefore, the value of (a + b)(a - b) is -1. It's important to pay attention to the sign (positive or negative) when performing subtractions, especially when dealing with different signs. A small mistake in the sign can lead to a completely different answer. Now, let's look back at the multiple-choice options provided in the problem. We had: (a) 1, (b) 2, (c) 3, and (d) 4. None of these options match our calculated result of -1. This indicates that there might be an error in the options provided in the question, or perhaps a deliberate attempt to test if we can identify that the correct answer is not listed. In a real-world scenario, this could happen in a test or an exam, and it's crucial to be confident in your calculations and recognize when the given options are incorrect. In this case, the correct answer is -1, which is not among the provided choices.

Conclusion: The Value of (a + b)(a - b) is -1

Alright guys, we've successfully navigated through this problem! By recognizing the difference of squares identity and applying it to the expression (a + b)(a - b), we were able to simplify the problem and easily substitute the given values of a² and b². Our calculations led us to the final result: (a + b)(a - b) = -1. This problem showcases the power of algebraic identities in simplifying complex expressions and making calculations more manageable. It also highlights the importance of double-checking your work and being confident in your answer, even if it doesn't match the given options. Remember, math is not just about finding the right answer; it's also about understanding the process and being able to apply the concepts in different situations. So, keep practicing, keep exploring, and keep that mathematical mindset sharp! You'll encounter similar problems in various contexts, and the skills you've honed here will definitely come in handy. Keep up the great work!