Missing Term In Factorization: Solve $100x^2 - 36$

Hey guys! Ever get stuck trying to figure out what's missing in a factorization problem? It can be a bit like solving a puzzle, but don't worry, we're here to break it down. Let's dive into a specific example: $100x^2 - 36 = 4(5x + ?)(5x - 3)$. We're going to walk through this step by step, so you'll not only understand the answer but also the why behind it. Understanding these concepts can really boost your math skills, and make tackling complex problems feel less daunting. So, let’s jump right in and make math a little less mysterious!

Understanding the Problem

In this section, we will dissect the problem to fully understand what is being asked and lay the groundwork for our solution. Let's begin by identifying the key components of the given equation: $100x^2 - 36 = 4(5x + ?)(5x - 3)$. The left side of the equation, $100x^2 - 36$, is a binomial, specifically a difference of squares. Recognizing this form is crucial because it allows us to apply a specific factoring pattern. On the right side, we have a factored expression with a missing term represented by the question mark (?). Our mission is to determine this missing term.

Recognizing the Difference of Squares

The expression $100x^2 - 36$ fits the pattern of a difference of squares, which is a fundamental concept in algebra. The general form of a difference of squares is $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. In our case, we can rewrite $100x^2$ as $(10x)^2$ and $36$ as $6^2$. Thus, our expression becomes $(10x)^2 - 6^2$. This recognition is a critical first step because it directs us towards a specific factoring strategy. Understanding this pattern not only helps in this problem but also in various other algebraic manipulations.

Analyzing the Factored Form

Now, let's examine the right side of the equation: $4(5x + ?)(5x - 3)$. We notice that the expression is partially factored. The presence of the term $(5x - 3)$ suggests that we are on the right track to factoring the difference of squares. The missing term inside the parenthesis $(5x + ?)$ is what we need to find. The coefficient 4 outside the parentheses indicates that a common factor might have been factored out initially. By carefully analyzing this form, we can deduce the relationship between the original expression and its factored form. This step is vital for connecting the factored form back to the original expression and identifying the missing piece of our puzzle.

Setting the Stage for the Solution

By understanding the difference of squares pattern and carefully analyzing the factored form, we have set the stage for solving the problem. We know that we need to find a term that, when multiplied with the given factors, will result in the original expression $100x^2 - 36$. This involves a bit of algebraic manipulation and careful attention to detail, but with our groundwork laid, we are well-prepared to find the missing term. This initial analysis is more than just a preliminary step; it’s the foundation upon which we build our solution, ensuring accuracy and understanding throughout the process.

Solving for the Missing Term

Alright, let's get down to business and find that missing term! To nail this, we're going to use our understanding of factoring and a little bit of algebraic savvy. Remember, the equation we're working with is $100x^2 - 36 = 4(5x + ?)(5x - 3)$. Our main goal here is to figure out what replaces that question mark, so the equation holds true. So, grab your thinking caps, guys, and let’s dive in!

Factoring out the Common Factor

The first thing we should do is look at the left side of the equation, $100x^2 - 36$. Notice anything? Both 100 and 36 are divisible by 4! That means we can factor out a 4 right off the bat. This simplifies the equation and makes it easier to work with. Factoring out the 4, we get: $4(25x^2 - 9)$. See how much cleaner that looks already? This step is super important because it aligns the left side of the equation more closely with the factored form on the right side, which is $4(5x + ?)(5x - 3)$. Factoring out common factors is like decluttering before a big project – it helps you see the essentials more clearly.

Applying the Difference of Squares

Now, let's focus on what's inside the parentheses: $25x^2 - 9$. Does this look familiar? It should! This is another difference of squares. Remember, the difference of squares pattern is $a^2 - b^2 = (a + b)(a - b)$. In our case, $25x^2$ is the same as $(5x)^2$, and 9 is $3^2$. So, we can apply the difference of squares pattern here. Factoring $25x^2 - 9$ gives us $(5x + 3)(5x - 3)$. This is a crucial step because it directly relates to the right side of our original equation. By recognizing and applying this pattern, we’re essentially unlocking the puzzle piece that will lead us to the solution. Plus, mastering the difference of squares pattern is a total game-changer for tackling all sorts of algebra problems.

Identifying the Missing Term

Okay, we're in the home stretch now! Let's put it all together. Our original equation was $100x^2 - 36 = 4(5x + ?)(5x - 3)$. We factored out a 4 from the left side, then applied the difference of squares, and we ended up with $4(25x^2 - 9) = 4(5x + 3)(5x - 3)$. Now, if we compare this to the right side of our original equation, which is $4(5x + ?)(5x - 3)$, it becomes crystal clear what the missing term is. Can you see it? The missing term is 3! We've successfully factored the expression and found the missing piece of the puzzle. This step is like the final reveal in a magic trick – it’s where everything comes together, and you see the complete picture. Pat yourselves on the back, guys, we nailed it!

Verifying the Solution

So, we've found our missing term, but how do we know for sure that we've got it right? That's where verification comes in! It's like double-checking your work to make sure everything adds up. We're going to plug our solution back into the original equation and make sure both sides are equal. This step is super important because it catches any sneaky errors and gives us confidence in our answer. Let’s break it down and make sure our math is solid!

Substituting the Value

First things first, let’s write down our original equation: $100x^2 - 36 = 4(5x + ?)(5x - 3)$. We figured out that the missing term is 3, so we're going to substitute 3 in place of the question mark. Our equation now looks like this: $100x^2 - 36 = 4(5x + 3)(5x - 3)$. This substitution is the key to verifying our solution. It transforms the equation into a concrete statement that we can either prove true or false. By plugging in our answer, we’re setting up a test to see if our solution holds water. This step is crucial because it directly links our solution back to the original problem, ensuring that we haven't gone astray in our calculations.

Expanding the Factored Form

Now, let's tackle the right side of the equation: $4(5x + 3)(5x - 3)$. To verify our solution, we need to expand this expression and see if it equals the left side, which is $100x^2 - 36$. We'll start by multiplying the two binomials inside the parentheses. Remember the difference of squares pattern? $(a + b)(a - b) = a^2 - b^2$. In our case, $a$ is $5x$ and $b$ is 3. So, $(5x + 3)(5x - 3)$ becomes $(5x)^2 - 3^2$, which simplifies to $25x^2 - 9$. Don't forget that 4 we factored out earlier! Now we multiply the entire expression by 4: $4(25x^2 - 9)$. Distributing the 4, we get $100x^2 - 36$. Bingo! This expansion is a critical step because it unravels the factored form and reveals its underlying structure. By carefully expanding the expression, we can directly compare it to the left side of our equation and confirm whether our solution is correct. This process not only verifies our answer but also reinforces our understanding of algebraic manipulation.

Comparing Both Sides

Alright, drumroll, please! Let's compare what we've got. On the left side of our original equation, we have $100x^2 - 36$. On the right side, after substituting and expanding, we also have $100x^2 - 36$. Guess what? They're the same! This means our solution is correct. The missing term is indeed 3. Guys, this step is the grand finale of our verification process. It’s the moment where we bring everything together and see the result of our hard work. When both sides of the equation match up perfectly, it’s like a symphony of mathematical harmony. Not only does this confirm our answer, but it also solidifies our understanding of the problem and the methods we used to solve it.

Conclusion

And there you have it! We successfully found the missing term in the factorization $100x^2 - 36 = 4(5x + ?)(5x - 3)$. By recognizing the difference of squares, factoring out common factors, and verifying our solution, we not only solved the problem but also reinforced some key algebraic concepts. Remember, guys, math is all about understanding the steps and why they work. Keep practicing, and you'll become factorization pros in no time! Understanding these principles will not only help you tackle similar problems but also build a strong foundation for more advanced mathematical concepts. So, keep up the great work, and happy problem-solving!