Calculating Expressions: A Math Problem Explained

Hey math enthusiasts! Let's dive into a cool algebra problem where we need to calculate the value of an expression. Specifically, we're going to tackle this one: (√2x-1)(√2x+1)-(√2x-1)²+(x-√2)², and we'll figure out its value when x equals -√3. Don't worry if it looks a bit intimidating at first – we'll break it down step by step to make it super clear and easy to follow. This is a fantastic opportunity to brush up on our algebra skills, especially those involving square roots and expanding expressions. Ready? Let's get started!

Unpacking the Expression: A Step-by-Step Approach

Alright, guys, the first thing we need to do is understand the expression itself. It's a mix of different algebraic terms, and the key to solving it is to systematically simplify each part. Let's break down each component: (√2x-1)(√2x+1) is the difference of squares, a common pattern in algebra. Then we have -(√2x-1)², which is the square of a binomial, and finally, we have (x-√2)², another square of a binomial. By simplifying these parts individually, we can then combine them to get our final result. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally, Addition and Subtraction (from left to right). This order is crucial to ensure that we get the right answer! Think of it like a recipe – you have to follow the steps in the correct order to get the perfect dish. In this case, our dish is the simplified value of the expression.

Now, let's look at the first term, (√2x-1)(√2x+1). Recognize this pattern? It’s the difference of squares! This can be expanded to (√2x)² - 1². The term (√2x)² is equal to 2x (the square root and the square cancel each other out), and 1² is, well, 1. So, this part simplifies to 2x - 1. Cool, right? Next up, -(√2x-1)². This part is a bit trickier because we have to remember the formula for squaring a binomial: (a - b)² = a² - 2ab + b². In our case, a = √2x and b = 1. So, (√2x-1)² becomes (√2x)² - 2*(√2x)1 + 1², which simplifies to 2x - 2√2x + 1. But remember, there's a negative sign in front, so we have to negate the entire result: -2x + 2√2x - 1. Finally, we have (x-√2)². Applying the same binomial square formula, this expands to x² - 2x*√2 + (√2)², which simplifies to x² - 2√2x + 2. Phew! That was a bit of work, but we’ve successfully simplified each part of the expression. Now, let’s bring it all together!

Bringing it All Together: Simplifying and Substituting

So, we have simplified each part of the expression. Now it's time to put it all together. Remember, the original expression was (√2x-1)(√2x+1)-(√2x-1)²+(x-√2)². Let's substitute the simplified forms we found in the previous step. We'll replace (√2x-1)(√2x+1) with 2x - 1, -(√2x-1)² with -2x + 2√2x - 1, and (x-√2)² with x² - 2√2x + 2. This gives us: 2x - 1 - 2x + 2√2x - 1 + x² - 2√2x + 2. Wow, that looks a bit less scary now, doesn’t it? Next up, we want to look for any like terms that we can combine. Like terms are those that have the same variable raised to the same power, or just constants.

Let’s start combining like terms. We have 2x and -2x, which cancel each other out. We also have 2√2x and -2√2x, which also cancel each other out. This leaves us with just x², and the constant terms -1, -1, and +2. Combining these, we get -1 -1 + 2 = 0. So, our simplified expression is x². Now we can easily move on to the substituting stage. We have finally boiled down the original expression into a much simpler form. So, our new expression looks way simpler, doesn't it? This step highlights the power of simplification: it can make complex problems much easier to handle. Isn't math amazing? It's like a puzzle where each step brings us closer to the solution. Now, let’s substitute x = -√3 into our simplified expression, x². This will allow us to find the specific numerical value of the expression for the given value of x. Let's do it!

Solving for x = -√3: The Final Calculation

Alright, guys, we’ve made it to the final step: solving the expression for x = -√3. After all that simplification, we're left with a super simple expression, x². All we have to do now is substitute -√3 in place of x. So, we have (-√3)². Remember that when you square a negative number, the result is positive. Squaring -√3 means multiplying -√3 by itself: (-√3) * (-√3). The square root of 3 multiplied by itself is 3. Since the two negatives cancel out, the result is positive. Thus, (-√3)² = 3. Ta-da! We've found our answer. The value of the original expression when x = -√3 is 3. Wasn't that awesome? We started with a complex-looking expression and, through careful simplification and substitution, arrived at a straightforward numerical answer. This example demonstrates how important it is to be patient and systematic when solving algebraic problems. Breaking down complex problems into smaller, more manageable steps makes the entire process much less intimidating and far more achievable.

Recap and Key Takeaways

Let's quickly recap everything we've done. We started with the expression (√2x-1)(√2x+1)-(√2x-1)²+(x-√2)², and our mission was to find its value when x = -√3. We began by simplifying each part of the expression using algebraic rules, such as the difference of squares and the binomial square formula. Then, we combined the simplified parts, looked for like terms to combine, and created a simplified expression: x². We then substituted x = -√3 into the simplified expression, and calculated the result, which was 3. This journey highlights several important mathematical concepts. First, understanding and applying algebraic formulas is crucial. Second, simplification is a powerful tool that transforms complex problems into simpler ones. Third, careful attention to the order of operations and the signs is essential for accuracy.

What are the key takeaways? Well, remember the patterns of the difference of squares, and the square of a binomial. These are your friends! The more you practice these, the faster and more comfortable you'll become with simplifying expressions. Always be mindful of the order of operations, and don't rush through the steps. Take your time, double-check your work, and don't be afraid to ask for help if you get stuck. Practice makes perfect, so keep practicing and experimenting with different types of algebraic problems. Each problem you solve is an opportunity to strengthen your skills and build your confidence. And most importantly, remember that math can be fun! With a bit of patience and the right approach, even the most complex expressions can be conquered. So, keep exploring the world of algebra, and enjoy the journey!

Further Exploration

Now that we've worked through this problem, you can try some similar exercises! For example, try to adapt this approach to solve other similar problems. You can also explore different values of x. What if x = 0 or x = 1? Also, try different expressions that involve square roots and binomials. Create your own problems and solve them. This is the best way to consolidate your skills and get better at math. Feel free to search online for more exercises and practice sheets. Websites like Khan Academy are perfect for extra practice and guidance. You could also try looking at more complex algebraic problems. Get creative! There are endless possibilities. The more you practice, the more confident and skilled you'll become. So, keep going, keep practicing, and enjoy the process of learning and discovery!